13.4: Series and Their Notations
- Use summation notation.
- Use the formula for the sum of the first \(n\) terms of an arithmetic series.
- Use the formula for the sum of the first \(n\) terms of a geometric series.
- Use the formula for the sum of an infinite geometric series.
- Solve annuity problems.
A couple decides to start a college fund for their daughter. They plan to invest \($50\) in the fund each month. The fund pays \(6\%\) annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in \(6\) years? In this section, we will learn how to answer this question. To do so, we need to consider the amount of money invested and the amount of interest earned.
Using Summation Notation
To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series . Consider, for example, the following series.
\(3+7+11+15+19+ \ldots \nonumber \)
The \(n^{th}\) partial sum of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation \(S_n\) represents the partial sum.
\[\begin{align} S_1 &= 3 \nonumber \\ S_2 &= 3+7=10 \nonumber \\ S_3 &= 3+7+11=21 \nonumber \\ S_4 &= 3+7+11+15 =36 \nonumber \end{align} \nonumber\]
Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma , \(\sum\), to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation , which is the number used to generate the first term in the series. The number above the sigma, called the upper limit of summation , is the number used to generate the last term in a series.
If we interpret the given notation, we see that it asks us to find the sum of the terms in the series \(a_k=2k\) for \(k=1\) through \(k=5\). We can begin by substituting the terms for \(k\) and listing out the terms of this series.
\[\begin{align}a_1 &=2(1)=2 \nonumber \\ a_2 &=2(2)=4 \nonumber \\ a_3 &= 2(3)=6 \nonumber \\ a_4 &= 2(4)=8 \nonumber \\ a_5 &= 2(5)=10 \nonumber \end{align} \nonumber\]
We can find the sum of the series by adding the terms:
\[\sum_{k=1}^{5}2k=2+4+6+8+10=30 \nonumber\]
The sum of the first \(n\) terms of a series can be expressed in summation notation as follows:
\[\sum_{k=1}^{n}a_k\]
This notation tells us to find the sum of \(a_k\) from \(k=1\) to \(k=n\).
\(k\) is called the index of summation, \(1\) is the lower limit of summation, and \(n\) is the upper limit of summation.
No. The lower limit of summation can be any number, but \(1\) is frequently used. We will look at examples with lower limits of summation other than \(1\).
- Identify the lower limit of summation.
- Identify the upper limit of summation.
- Substitute each value of \(k\) from the lower limit to the upper limit into the formula.
- Add to find the sum.
Evaluate
\(\sum_{k=3}^{7}k^2\)
Solution
According to the notation, the lower limit of summation is \(3\) and the upper limit is \(7\). So we need to find the sum of \(k^2\) from \(k=3\) to \(k=7\). We find the terms of the series by substituting \(k=3, 4, 5, 6,\) and \(7\) into the function \(k^2\). We add the terms to find the sum.
\[\begin{align}\sum_{k=3}^{7}k^2 &= 3^2+4^2+5^2+6^2+7^2 \nonumber \\[4pt] &= 9+16+25+36+49 \nonumber \\[4pt] &= 135 \nonumber \end{align} \nonumber\]
Evaluate
\(\sum_{k=2}^{5}(3k–1)\)
- Answer
-
\(38\)
Using the Formula for Arithmetic Series
Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference , \(d\). The sum of the terms of an arithmetic sequence is called an arithmetic series . We can write the sum of the first \(n\) terms of an arithmetic series as:
\[S_n=a_1+(a_1+d)+(a_1+2d)+...+(a_n–d)+a_n. \nonumber \]
We can also reverse the order of the terms and write the sum as
\[S_n=a_n+(a_n–d)+(a_n–2d)+...+(a_1+d)+a_1. \nonumber\]
If we add these two expressions for the sum of the first \(n\) terms of an arithmetic series, we can derive a formula for the sum of the first \(n\) terms of any arithmetic series.
\[S_n=a_1+(a_1+d)+(a_1+2d)+...+(a_n–d)+a_n \nonumber\]
\[\underline{+S_n=a_n+(a_n–d)+(a_n–2d)+...+(a_1+d)+a_1} \nonumber\]
\[2S_n=(a_1+a_n)+(a_1+a_n)+...+(a_1+a_n) \nonumber\]
Because there are \(n\) terms in the series, we can simplify this sum to
\[2S_n=n(a_1+a_n). \nonumber \]
We divide by \(2\) to find the formula for the sum of the first \(n\) terms of an arithmetic series.
\[S_n=\dfrac{n(a_1+a_n)}{2} \nonumber \]
An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of the first \(n\) terms of an arithmetic sequence is
\[S_n=\dfrac{n(a_1+a_n)}{2}\]
- Identify \(a_1\) and \(a_n\).
- Determine \(n\).
- Substitute values for \(a_1\), \(a_n\), and \(n\) into the formula \(S_n=\dfrac{n(a_1+a_n)}{2}\).
- Simplify to find \(S_n\).
Find the sum of each arithmetic series.
- \(5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32\)
- \(20 + 15 + 10 +…+ −50\)
- \(\sum_{k=1}^{12}3k−8\)
Solution
- We are given \(a_1=5\) and \(a_n=32\).
Count the number of terms in the sequence to find \(n=10\).
Substitute values for \(a_1\), \(a_n\), and \(n\) into the formula and simplify.
\[\begin{align*} S_n&=\dfrac{n(a_1+a_n)}{2}\\ S_{10}&=\dfrac{10(5+32)}{2}\\ &=185 \end{align*}\]
- We are given \(a_1=20\) and \(a_n=−50\).
Use the formula for the general term of an arithmetic sequence to find \(n\).
\[\begin{align*} a_n&=a_1+(n-1)d\\ -50&=20+(n-1)(-5)\\ -70&=(n-1)(-5)\\ 14&=n-1\\ 15&=n \end{align*}\]
Substitute values for \(a_1\), \(a_n\), \(n\) into the formula and simplify.
\[\begin{align*} S_n&=\dfrac{n(a_1+a_n)}{2}\\ S_{15}&=\dfrac{15(20-50)}{2}\\ &=-225 \end{align*}\]
- To find \(a_1\), substitute \(k=1\) into the given explicit formula.
\[\begin{align*} a_k&=3k-8\\ a_1&=3(1)-8\\ &=-5 \end{align*}\]
We are given that \(n=12\). To find \(a_12\), substitute \(k=12\) into the given explicit formula.
\[\begin{align*} a_k&=3k-8\\ a_{12}&=3(12)-8\\ &=28 \end{align*}\]
Substitute values for \(a_1\), \(a_n\), and \(n\) into the formula and simplify.
\[\begin{align*} S_n&=\dfrac{n(a_1+a_n)}{2}\\ S_{12}&=\dfrac{12(-5+28)}{2}\\ &=138 \end{align*}\]
Use the formula to find the sum of each arithmetic series.
\(1.4 + 1.6 + 1.8 + 2.0 + 2.2 + 2.4 + 2.6 + 2.8 + 3.0 + 3.2 + 3.4\)
- Answer
-
\(26.4\)
\(13 + 21 + 29 + …+ 69\)
- Answer
-
\(328\)
\(\sum_{k=1}{10}5−6k\)
- Answer
-
\(−280\)
On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After \(8\) weeks, what will be the total number of miles she has walked?
Solution
This problem can be modeled by an arithmetic series with \(a_1=\dfrac{1}{2}\) and \(d=\dfrac{1}{4}\). We are looking for the total number of miles walked after \(8\) weeks, so we know that \(n=8\), and we are looking for \(S_8\). To find \(a_8\), we can use the explicit formula for an arithmetic sequence.
\[\begin{align*} a_n&=a_1+d(n-1)\\ a_8&=\dfrac{1}{2}+\dfrac{1}{4}(8-1)\\ &=\dfrac{9}{4} \end{align*}\]
We can now use the formula for arithmetic series.
\[\begin{align*} S_n&=\dfrac{n(a_1+a_n)}{2}\\ S_8&=\dfrac{8\left(\dfrac{1}{2}+\dfrac{9}{4}\right)}{2}\\ &=11 \end{align*}\]
She will have walked a total of \(11\) miles.
A man earns \($100\) in the first week of June. Each week, he earns \($12.50\) more than the previous week. After \(12\) weeks, how much has he earned?
- Answer
-
\($2,025\)
Using the Formula for Geometric Series
Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series . Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio , \(r\). We can write the sum of the first \(n\) terms of a geometric series as
\(S_n=a_1+ra_1+r^2a_1+...+r^{n–1}a_1\).
Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first \(n\) terms of a geometric series. We will begin by multiplying both sides of the equation by \(r\).
\(rS_n=ra_1+r^2a_1+r^3a_1+...+r^na_1\)
Next, we subtract this equation from the original equation.
\[\begin{align*} S_n&=a_1+ra_1+r^2a_1+...+r^{n-1}a_1\\ \underline{-rS_n}&=\underline{-(ra_1+r^2a_1+r^3a_1+...+r^na_1)}\\ (1-r)S_n&=a_1-r^na_1 \end{align*}\]
Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for \(S_n\), divide both sides by \((1−r)\).
\(S_n=\dfrac{a_1(1−r^n)}{1−r}\) \(r≠1\)
A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the first \(n\) terms of a geometric sequence is represented as
\(S_n=\dfrac{a_1(1−r^n)}{1−r}\) \(r≠1\)
- Identify \(a_1\), \(r\), and \(n\).
- Substitute values for \(a_1\), \(r\), and \(n\) into the formula \(S_n=\dfrac{a_1(1−r^n)}{1−r}\).
- Simplify to find \(S_n\).
Use the formula to find the indicated partial sum of each geometric series.
- \(S_{11}\) for the series \(8 + -4 + 2 + …\)
- \(\sum_{ 6}^{k=1}3⋅2k\)
Solution
- \(a_1=8\), and we are given that \(n=11\).
We can find \(r\) by dividing the second term of the series by the first.
\(r=\dfrac{−4}{8}=−\dfrac{1}{2}\)
Substitute values for \(a_1\), \(r\), and \(n\) into the formula and simplify.
\[\begin{align*} S_n&=\dfrac{a_1(1-r^n)}{1-r}\\ S_{11}&=\dfrac{8\left(1-{\left(-\dfrac{1}{2}\right)}^{11}\right)}{1-\left(-\dfrac{1}{2}\right)}\\ &\approx 5.336 \end{align*}\]
- Find \(a_1\) by substituting \(k=1\) into the given explicit formula.
\(a_1=3⋅2^1=6\)
We can see from the given explicit formula that \(r=2\). The upper limit of summation is \(6\), so \(n=6\).
Substitute values for \(a_1\), \(r\), and \(n\) into the formula, and simplify.
\[\begin{align*} S_n&=\dfrac{a_1(1-r^n)}{1-r}\\ S_6&=\dfrac{6(1-2^6)}{1-2}\\ &=378 \end{align*}\]
Use the formula to find the indicated partial sum of each geometric series.
\(S_{20}\) for the series \(1,000 + 500 + 250 + …\)
- Answer
-
\(≈2,000.00\)
\(\sum_{k=1}^{8}3^k\)
- Answer
-
\(9,840\)
At a new job, an employee’s starting salary is \($26,750\). He receives a \(1.6\%\) annual raise. Find his total earnings at the end of \(5\) years.
Solution
The problem can be represented by a geometric series with \(a_1=26,750\); \(n=5\); and \(r=1.016\). Substitute values for \(a_1\), \(r\), and \(n\) into the formula and simplify to find the total amount earned at the end of \(5\) years.
\[\begin{align*} S_n&=\dfrac{a_1(1-r^n)}{1-r}\\ S_5&=\dfrac{26,750(1-{1.016}^5)}{1-1.016}\\ &\approx 138,099.03 \end{align*}\]
He will have earned a total of \($138,099.03\) by the end of \(5\) years.
At a new job, an employee’s starting salary is \($32,100\). She receives a \(2\%\) annual raise. How much will she have earned by the end of \(8\) years?
- Answer
-
\($275,513.31\)
Using the Formula for the Sum of an Infinite Geometric Series
Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first \(n\) terms. An infinite series is the sum of the terms of an infinite sequence . An example of an infinite series is \(2+4+6+8+...\)
This series can also be written in summation notation as \(\sum_{k=1}^{\infty}2k\), where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges .
Determining Whether the Sum of an Infinite Geometric Series is Defined
If the terms of an infinite geometric series approach \(0\), the sum of an infinite geometric series can be defined. The terms in this series approach \(0\):
\(1+0.2+0.04+0.008+0.0016+...\)
The common ratio \(r = 0.2\). As \(n\) gets very large, the values of \(r^n\) get very small and approach \(0\). Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to \(0\), the sum of the terms approaches a finite value. The terms of any infinite geometric series with \(−1<r<1\) approach \(0\); the sum of a geometric series is defined when \(−1<r<1\).
The sum of an infinite series is defined if the series is geometric and \(−1<r<1\).
- Find the ratio of the second term to the first term.
- Find the ratio of the third term to the second term.
- Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.
- If a common ratio, \(r\), was found in step 3, check to see if \(−1<r<1\). If so, the sum is defined. If not, the sum is not defined.
Determine whether the sum of each infinite series is defined.
- \(12 + 8 + 4 + …\)
- \(\dfrac{3}{4}+\dfrac{1}{2}+\dfrac{1}{3}+...\)
- \(\sum_{k=1}^{\infty}27⋅{(\dfrac{1}{3})}^k\)
- \(\sum_{k=1}^{\infty}5k\)
Solution
- The ratio of the second term to the first is \(\dfrac{2}{3}\), which is not the same as the ratio of the third term to the second, \(\dfrac{1}{2}\).The series is not geometric.
- The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of \(\dfrac{2}{3}\). The sum of the infinite series is defined.
- The given formula is exponential with a base of \(\dfrac{1}{3}\); the series is geometric with a common ratio of \(\dfrac{1}{3}\). The sum of the infinite series is defined.
- The given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot yield a finite sum.
Determine whether the sum of the infinite series is defined.
\(\dfrac{1}{3}+\dfrac{1}{2}+\dfrac{3}{4}+\dfrac{9}{8}+...\)
- Answer
-
The sum is defined. It is geometric.
\(24+(−12)+6+(−3)+...\)
- Answer
-
The sum of the infinite series is defined.
\(\sum_{k=1}^{\infty}15⋅{(–0.3)}^k\)
- Answer
-
The sum of the infinite series is defined.
Finding Sums of Infinite Series
When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first \(n\) terms of a geometric series.
\[S_n=\dfrac{a_1(1−r^n)}{1−r}\]
We will examine an infinite series with \(r=\dfrac{1}{2}\). What happens to \(r^n\) as \(n\) increases?
\[\begin{align*} \left(\dfrac{1}{2}\right)^2&=\dfrac{1}{4}\\ \left(\dfrac{1}{2}\right)^3&=\dfrac{1}{8}\\ \left(\dfrac{1}{2}\right)^4&=\dfrac{1}{16} \end{align*}\]
The value of \(r^n\) decreases rapidly. What happens for greater values of \(n\)?
\[\begin{align*} {\left(\dfrac{1}{2}\right)}^{10} &= \dfrac{1}{1,024} \\ {\left(\dfrac{1}{2}\right)}^{20} &= \dfrac{1}{1,048,576} \\ {\left(\dfrac{1}{2}\right)}^{30} &= \dfrac{1}{1,073,741,824} \end{align*}\]
As \(n\) gets very large, \(r^n\) gets very small. We say that, as \(n\) increases without bound, \(r^n\) approaches 0. As \(r^n\) approaches \(0\), \(1\), \(−r^n\) approaches \(1\). When this happens, the numerator approaches \(a_1\). This give us a formula for the sum of an infinite geometric series.
The formula for the sum of an infinite geometric series with \(−1<r<1\) is
\[S=\dfrac{a_1}{1−r}\]
- Identify \(a_1\) and \(r\).
- Confirm that \(–1<r<1\).
- Substitute values for \(a_1\) and \(r\) into the formula, \(S=\dfrac{a_1}{1−r}\).
- Simplify to find \(S\).
Find the sum, if it exists, for the following:
- \(10+9+8+7+…\)
- \(248.6+99.44+39.776 + …\)
- \(\sum_{k=1}^{\infty}4,374⋅{(–\dfrac{1}{3})}^{k–1}\)
- \(\sum_{k=1}^{\infty}\dfrac{1}{9}⋅{(\dfrac{4}{3})}^k\)
Solution
- There is not a constant ratio; the series is not geometric.
- There is a constant ratio; the series is geometric. \(a_1=248.6\) and \(r=\dfrac{99.44}{248.6}=0.4\), so the sum exists. Substitute \(a_1=248.6\) and \(r=0.4\) into the formula and simplify to find the sum:
\[ \begin{align*} S&=\dfrac{a_1}{1−r} \\[4pt] &=\dfrac{248.6}{1−0.4} \\[4pt] &=414.\overline{3} \end{align*}\]
- The formula is exponential, so the series is geometric with \(r=–\dfrac{1}{3}\). Find \(a_1\) by substituting \(k=1\) into the given explicit formula:
\(a_1=4,374⋅{(–\dfrac{1}{3})}^{1–1}=4,374\)
Substitute \(a_1=4,374\) and \(r=−\dfrac{1}{3}\) into the formula, and simplify to find the sum:
\[ \begin{align*} S&=\dfrac{a_1}{1−r} \\[4pt] &= \dfrac{4,374}{1−\left(−\dfrac{1}{3}\right)} \\[4pt] &= 3,280.5 \end{align*}\]
- The formula is exponential, so the series is geometric, but \(r>1\). The sum does not exist.
Find an equivalent fraction for the repeating decimal \(0.\overline{3}\)
Solution
We notice the repeating decimal \(0.\overline{3}=0.333...\) so we can rewrite the repeating decimal as a sum of terms.
\(0.\overline{3}=0.3+0.03+0.003+...\)
Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to \(0.1\) in the second term, and the second term multiplied to \(0.1\) in the third term.
Notice the pattern; we multiply each consecutive term by a common ratio of \(0.1\) starting with the first term of \(0.3\). So, substituting into our formula for an infinite geometric sum, we have
\(S_n=\dfrac{a_1}{1−r}=\dfrac{0.3}{1−0.1}=\dfrac{0.3}{0.9}=\dfrac{1}{3}\).
Find the sum, if it exists.
\(2+23+29+...\)
- Answer
-
\(3\)
\(\sum_{k=1}^{\infty}0.76k+1\)
- Answer
-
The series is not geometric.
\(\sum_{k=1}^{\infty}{\left(−\dfrac{3}{8}\right)}^k\)
- Answer
-
\(−\dfrac{3}{11}\)
Solving Annuity Problems
At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the couple invests \($50\) each month. This is the value of the initial deposit. The account paid \(6\%\) annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the \(6\%\) annual percentage interest (APR) rate by \(12\). So the monthly interest rate is \(0.5\%\). We can multiply the amount in the account each month by \(100.5\%\) to find the value of the account after interest has been added.
We can find the value of the annuity right after the last deposit by using a geometric series with \(a_1=50\) and \(r=100.5%=1.005\). After the first deposit, the value of the annuity will be \($50\). Let us see if we can determine the amount in the college fund and the interest earned.
We can find the value of the annuity after nn deposits using the formula for the sum of the first nn terms of a geometric series. In \(6\) years, there are \(72\) months, so \(n=72\). We can substitute \(a_1=50\), \(r=1.005\), and \(n=72\) into the formula, and simplify to find the value of the annuity after 6 years.
\(S_{72}=\dfrac{50(1−{1.005}^{72})}{1−1.005}≈4,320.44\)
After the last deposit, the couple will have a total of \($4,320.44\) in the account. Notice, the couple made \(72\) payments of \($50\) each for a total of \(72(50) = $3,600\). This means that because of the annuity, the couple earned \($720.44\) interest in their college fund.
- Determine \(a_1\), the value of the initial deposit.
- Determine \(n\), the number of deposits.
-
Determine \(r\).
- Divide the annual interest rate by the number of times per year that interest is compounded.
- Add 1 to this amount to find \(r\).
- Substitute values for \(a_1\), \(r\), and \(n\) into the formula for the sum of the first nn terms of a geometric series, \(S_n=\dfrac{a_1(1–r^n)}{1–r}\).
- Simplify to find \(S_n\), the value of the annuity after \(n\) deposits.
A deposit of \($100\) is placed into a college fund at the beginning of every month for \(10\) years. The fund earns \(9\%\) annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?
Solution
The value of the initial deposit is \($100\), so \(a_1=100\). A total of \(120\) monthly deposits are made in the \(10\) years, so \(n=120\). To find \(r\), divide the annual interest rate by \(12\) to find the monthly interest rate and add \(1\) to represent the new monthly deposit.
\(r=1+\dfrac{0.09}{12}=1.0075\)
Substitute \(a_1=100\), \(r=1.0075\), and \(n=120\) into the formula for the sum of the first \(n\) terms of a geometric series, and simplify to find the value of the annuity.
\(S_{120}=\dfrac{100(1−{1.0075}^{120})}{1−1.0075}≈19,351.43\)
So the account has $19,351.43 after the last deposit is made.
At the beginning of each month, \($200\) is deposited into a retirement fund. The fund earns \(6\%\) annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for \(10\) years?
- Answer
-
\($92,408.18\)
Access these online resources for additional instruction and practice with series.
Key Equations
| sum of the first \(n\) terms of an arithmetic series | \(S_n=\dfrac{n(a_1+a_n)}{2}\) |
| sum of the first \(n\) terms of a geometric series | \(S_n=\dfrac{a_1(1−r^n)}{1−r}\)⋅ \(r≠1\) |
| sum of an infinite geometric series with \(–1<r< 1\) | \(S_n=\dfrac{a_1}{1−r}\)⋅ \(r≠1\) |
Key Concepts
- The sum of the terms in a sequence is called a series.
- A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See Example \(\PageIndex{1}\).
- The sum of the terms in an arithmetic sequence is called an arithmetic series.
- The sum of the first \(n\) terms of an arithmetic series can be found using a formula. See Example \(\PageIndex{2}\) and Example \(\PageIndex{3}\).
- The sum of the terms in a geometric sequence is called a geometric series.
- The sum of the first \(n\) terms of a geometric series can be found using a formula. See Example \(\PageIndex{4}\) and Example \(\PageIndex{5}\).
- The sum of an infinite series exists if the series is geometric with \(–1<r<1\).
- If the sum of an infinite series exists, it can be found using a formula. See Example \(\PageIndex{6}\), Example \(\PageIndex{7}\) , and Example \(\PageIndex{8}\).
- An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See Example \(\PageIndex{9}\).


30.5: Further Reading for Scholarly Works of Incidents in the Life of a Slave Girl (Group Anthology Contribution 2019)
This is an annotated bibliography of several works pertaining to Incidents in the Life of a Slave Girl . In previous versions of the online anthology, there were no annotated bibliographies for this text. We feel it is important to include these literary criticisms because they help us see the many themes in Incidents. We hope future students, as well as instructors, can find this bibliography useful!
Garfield, Deborah M. “Speech, Listening and Female Sexuality in Incidents in the Life of a Slave Girl.” Arizona Quarterly: A Journal of American Literature, Culture, and Theory, vol. 50, no. 2, 1994, pp. 19-49. Project Muse, doi:10.1353/arq.1994.0012. 8 November 2019.
Garfield connects Jacobs’s Incidents in the Life of a Slave Girl to a much wider literary and oral tradition of slave narratives at the time. Jacobs wrote her ground-breaking text during a time when slaves started speaking out to white audiences about their time in slavery. Garfield delves briefly into the history of slaves trying to navigate this new sphere. They found that oratory narratives connected more strongly with the audience than written texts. Thus, many freed slaves started touring, giving lectures on what their life had been like as enslaved people. White folk resonated with these stories despite their horror. However there was one brand of slave narrative that people did not take to well: the female narrative. Female slaves often experienced horrific sexual assault. Their narratives were deemed corrupting for innocent white women. Therefore, enslaved peoples were censored in telling their authentic stories. They had to find alternate ways to get their messages across. Some took on oratory styles that were more traditionally “male.” Others, like Jacobs, resigned to trading out the more effective speech for a written text.
Garfield highlights may instances where Jacobs utilizes stylistic elements to soften the blow of her trauma for a white, female audience. Even though Linda experiences sexual assault in the story, Jacobs is very careful at how she portrays this reality. Dr. Flint, Linda’s main abuser, is oftentimes seen in the text verbally assaulting Linda even though the reader knows that much of her abuse is physical. Garfield notes that Jacobs purposefully did this to shelter her readers. These dynamics parallel physical rape; Linda must stand passively and take the aggression of Dr. Flint. Thus, the reader still experiences Linda’s reality, but in a way that is more palatable. Garfield highlights the gender politics in Incidents in the Life of a Slave Girl and connects them to the events that were happening at the time. This context supports a deeper understanding of the characters and dynamics working in and on the text.
Gomaa, Sally. “Writing to ‘Virtuous’ and ‘Gentle’ Readers: The Problem of Pain in Harriet Jacobs’s Incidents and Harriet Wilson’s Sketches .” African American Review , vol. 43, no. 2/3, 2009, pp. 371-81 Accessed 9 Nov. 2019.
Gomaa goes into detail about how slave bodies were treated during their fight for freedom. Abolitionists required/pressured freed slaves to show people their physical scars so they could gain sympathy and further the rebellion. She talks about “sentimentalizing pain” in various slave narratives, including Incidents in the Life of a Slave Girl. Sentimentalizing pain is the idea that one needs to make the reader live their pain through details of their lives or by comparing it to another kind of pain. The idea is that this preludes getting sympathy for said pain. She brings up how Jacobs refers to the body as “home,” creating a metaphor for readers to understand better and make a statement. Gomaa goes into detail about how Jacobs accomplishes these ideas throughout Incidents. It’s important to understand this idea because it helps us see the reasons behind Jacobs’ writing choices in Incidents.
Larson, Jennifer. “Converting Passive Womanhood to Active Sisterhood: Agency, Power, and Subversion in Harriet Jacobs’s Incidents in the Life of a Slave Girl .” Women’s Studies , vol. 35, no. 8, Dec. 2006, pp. 739-56 Accessed 8 Nov. 2019.
In this literary criticism, Larson details what the “Cult of True Womanhood” is and how it connects with Incidents . She talks about how Jacobs attacks the ideals that are spread by them, particularly about purity and submissiveness. Purity and religious devotion were important in the 1800s, dictating how women needed to act. Larson details throughout this piece how Jacobs puts forth a call to action for womanhood. She suggests that Jacobs was showing women that community, or sisterhood, was more important than being domestic. This piece is important to Incidents in the Life of a Slave Girl because it shows how it is a feminist text. It is important for readers to see how Incidents can be a feminist text beyond having a persevering female lead. We feel that this piece might help readers see that.
Wesley, Marilyn C. “A Woman’s Place: The Politics of Space in Harriet Jacobs’ Incidents in the Life of a Slave Girl. ” Womens Studies, vol. 26, no. 1, 1997, pp. 59-72. WorldCat Discovery . 8 November 2019.
Wesley takes an in-depth look into the underlying feminine narrative in Incidents in the Life of a Slave Girl. She explains the narrative through the lens of the travel text and makes a multitude of comparisons to works of Frederick Douglass . While Douglass writes about masculine characters that fight for their freedom with physical strength and brutality, Linda is more constrained in her fight. Linda fought with her body, by getting pregnant with her white lover to upset Dr. Flint. While men in traditional slave and travel texts go off into the world and have an adventure, Linda shuts herself up in her grandmother’s tiny crawlspace. Here Wesley talks about gendered space. Males slaves often run off to the north and fight for their freedom there.  From her hole in the South, Linda starts her path towards freedom. She is able to gain power in that space as she has Dr. Flint chasing her fake letters around the North while she sits in her grandmother’s house. Westley notes how the later part of the narrative shifts, as Linda begins her travels around the North. The residue of her trauma as a slave is seen as she maneuvers the free world. Thus, because she knows that the world can still overpower her, she is not truly free. Finally Linda’s employer buys her freedom. While this should have come to relief for Linda, she still feels that she is controlled by the institution of slavery. While the protagonists of Douglass’s stories gain their own freedom, Linda eventually has it given to her. Wesley highlights how the feminine aspect of Incidents in the Life of a Slave Girl changes the narrative for the protagonist.
Whitsitt, Novian. “Reading Between the Lines: The Black Cultural Tradition of Masking in Harriet Jacobs’s Incidents in the Life of a Slave Girl .” Frontiers: A Journal of Women Studies, vol. 31, no. 1, 2010, pp. 73-88. Academic Search Premier . Accessed 8 Nov. 2019.
Whitsitt delves into the reliability and double voice of this slave narrative. Whitsitt suggests that Jacobs’ use of double voice allows her to tell the truth while appealing to a majority of her audience (Northern white women). This use of double voice affects the slave narrative in discrete ways, challenging the reader to “read between the lines” in order to find out the whole truth about slavery. This essay also discusses Jacobs’ use of feminine ideals and Christianity, explaining how they are used to gain trust from the reader base. This is an extremely important reading to bring to light due to the way Incidents In The Life Of A Slave Girl is received. Typically, the narrative is read as complete truth with few critics questioning the reliability. By bringing in the question of how reliable it is, Whitsitt shows us how even this slave narrative is affected by white power.


1.5: Low-temperature and sous-vide cooking in the pastry shop
Sous-vide cooking is about immersing a food item in a precisely controlled water bath, where the temperature of the water is the same as the target temperature of the food being cooked. Food is placed in a food-grade plastic bag and vacuum-sealed before going into the water bath. Temperatures will vary depending on desired end result. This allows the water in the bath to transfer heat into the food while preventing the water from coming into direct contact with it. This means the water does not chemically interact with the food: the flavours of the food remain stronger, because the water is unable to dissolve or carry away any compounds in the food (Figure 4).
Applications for the bakeshop
Sous-vide fruits and vegetables
Cooking vegetable and fruits sous-vide is a great way to tenderize them without losing as many of the vitamins and minerals that are normally lost through blanching or steaming. Fruits can also be infused with liquid when cooked at lower temperatures by adding liquid to the bag. Sous-vide helps preserve the nutrients present in fruits and vegetables by not cooking them above the temperatures that cause the cell walls to fully break down. This allows them to tenderize without losing all their structure. The bag also helps to catch any nutrients that do come out of the vegetable.
While time and temperature do not factor into safety for fruits and vegetables, they do have a unique effect on their structure. There are two components in fruits and vegetables that make them crisp: pectin and starch. Pectin, which is a gelling agent commonly used in jams and jellies for structure, breaks down at 83ºC (183ºF) at a slower rate than the starch cells do. In many cases this allows for more tender fruits and vegetables that have a unique texture to them.
Custards
The term custard spans so many possible ingredients and techniques that it is most useful to think of a custard as simply a particular texture and mouth feel. Custards have been made for centuries by lightly cooking a blend of eggs, milk, and heavy cream, but modernist chefs have invented myriad ways to make custards.
Using the sous-vide method to prepare crème anglaise, curds, ice cream bases, custard bases, sabayons, and dulce de leche is possible. The technique offers greater consistency and more control over the texture, which can range from airy, typical of a sabayon, to dense, as in a posset. For custards, eggs will be properly cooked at 82°C (180ºF), so if the water bath is set to this temperature, no overcooking can happen. The one constant among custards is the use of plenty of fat, which not only provides that distinctive mouth feel but also makes custard an excellent carrier of fat-soluble flavours and aromas. Lighter varieties of custard, prepared sous-vide style and cooled, can be aerated in a whipping siphon into smooth, creamy foams.
Other applications for vacuum-seal processes
Fruit compression
Vacuum-compressing fruits and vegetables is a popular modern technique that can give many plant foods an attractive, translucent appearance (as shown in the watermelon in Figure 5) and a pleasant, surprising texture. This technique exploits the ability of a vacuum chamber to reduce surrounding pressure, which causes air and moisture within the plant tissue to rapidly expand and rupture the structures within the food. When the surrounding pressure is restored to a normal level, the labyrinth of air-filled spaces collapses. As a result, light tends to pass through the food rather than being scattered and diffused, which is why vacuum-compressed plant foods appear translucent. Causing the porous structure of a plant food to collapse also imparts a somewhat dense, toothsome texture that can give a familiar ingredient, such as watermelon, an entirely new appeal.
Infusions
When adding liquids, the vacuum-seal process creates a rapid infusion—especially with more porous foods (such as adding spices to cream or herbs to melon). This can add flavour and texture in a shorter time than traditional infusions.
Media Attributions
- Sous Vide Cooking by Derek © CC BY-ND (Attribution NoDerivatives)
- WD-50 (7th Course) by Peter Dillon © CC BY (Attribution)


1.12: Beatriz, Countess of Dia
Our listening example from the troubadour tradition comes from the most famous of the female troubadors ( trobairitz ), Beatriz of Dia. Much uncertainty surrounds the historical records of the troubadors, but here is some brief information to help you get a sense of her place in the world of medieval secular music.
Introduction
The Comtessa de Dia , probably named Beatritz or Beatriz (fl. c. 1175), was a trobairitz (female troubadour).
She is only known as the comtessa de Dia in contemporary documents, but was almost certainly named Beatriz and likely the daughter of Count Isoard II of Diá (a town northeast of Montelimar in southern France). According to her vida , she was married to William of Poitiers, but was in love with and sang about Raimbaut of Orange (1146-1173). It has been hypothesised that she was in fact married to Guillem’s son, Ademar de Peiteus, whose wife’s name was Philippa de Fay, and that her real lover was Raimbaut de Vaqueiras.
Beatrice’s poems were often set to the music of a flute. Five of her works survive, including 4 cansos and 1 tenson. Scholars have debated whether or not Comtessa authored Amics, en greu consirier, a tenso typically attributed to Raimbaut d’Aurenga. One reason for this is due to the similarities between this composition and her own Estat ai en greu consirier. A second reason references the words in her vida, Et enamoret se d’En Rambaut d’ Ashley, e fez de lui mantas bonas cansos (and she fell in love with Sir Raimbaut d’Aurenga, and made about him many good cansos) .
Her song A chantar m’er de so qu’eu no volria in the Occitan language is the only canso by a trobairitz to survive with its music intact. The music to A chantar is found only in Le manuscript di roi, a collection of songs copied around 1270 for Charles of Anjou, the brother of Louis IX.
Her extant poems are:
- Ab joi et ab joven m’apais
- A chantar m’er de so qu’ieu non volria
- Estât ai en greu cossirier
- Fin ioi me don’alegranssa
Typical subject matter used by Comtessa de Dia in her lyrics includes optimism, praise of herself and her love, as well as betrayal. In A chantar, Comtessa plays the part of a betrayed lover, and despite the fact she has been betrayed, continues to defend and praise herself. In Fin ioi me don’alegranssa, however, Comtessa makes fun of the lausengier, a person known for gossiping, comparing those who gossip to a “cloud that obscures the sun.” In writing style, Comtessa uses a process known as coblas singulars in A chantar, repeating the same rhyme scheme in each strophe, but changing the a rhyme each time. Ab ioi, on the other hand, uses coblas doblas, with a rhyme scheme of ab’ ab’ b’ aab’. A chantar uses some of the motifs of Idyll II of Theocritus.
Listen: A chantar m’er
The only existing song by a trobairitz that survives with music.
An audio element has been excluded from this version of the text. You can listen to it online here: http://pb.libretexts.org/map/?p=68
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Beatritz de Dia. Provided by : Wikipedia . Located at : http://en.Wikipedia.org/wiki/Beatritz_de_Dia . License : CC BY-SA: Attribution-ShareAlike


9.7: Psychological Trauma of Violence Against Women
By the end of this section, you will be able to:
- Discuss the definition of psychologic abuse and the role of the nurse
- Identify clinical manifestations of IPV
- Describe programs to assist women who have been victimized
Safety is a basic human need and always receives priority in patient care. Nurses typically use Maslow’s Hierarchy of Needs (Figure 9.4) to prioritize urgent patient needs, with the bottom two rows of the pyramid receiving top priority. Safety is intertwined with basic physiologic needs.
Psychologic abuse violates a person’s feelings of safety and self-worth, inhibiting their basic human needs. Because their basic human needs are not met, the overall health of the victim declines. Abuse eventually affects a woman’s reproductive and mental health. Violence also destroys relationships and decreases the victim’s confidence. Nurses can lobby organizations and lawmakers to help stop violence against women .
Psychologic Abuse
The type of abuse that can include harassment ; verbal abuse such as name calling, degradation, and blaming; threats; stalking; and isolation with the intent to attack the person’s self-esteem and self-worth, making the person reliant on their abuser is psychologic abuse (National Network to End Domestic Violence, 2021). Abusers often isolate their partner from family members and friends. They may deprive them of essentials, such as food, money, transportation, and access to health care (National Network to End Domestic Violence, 2021). The abuser thrives on control and manipulation. The abuser can be a coworker, boss, family member, intimate partner, or acquaintance. Name-calling, insults, belittling or threats can lead to long-term psychologic harm to the person (National Network to End Domestic Violence, 2021). Signs of psychologic abuse may include being withdrawn or isolated, emotional reactions that are not consistent with the situation, or other behavior outside the norm for that person (Washington State Department of Social and Health Services, n.d.).
The nurse who suspects a patient is experiencing psychologic abuse should take steps similar to those for suspected physical abuse . The patient should be given resources and referrals. The nurse should also educate the patient that psychologic abuse is as significant a health concern as physical abuse.
Clinical Manifestations of Psychologic Abuse
Psychologic abuse leads to outward manifestations of the abuse, such as mental health problems, cognitive issues, reproductive issues, and relationship problems. Many times, abusers will threaten to hurt themselves if the person decides to end the relationship (National Network to End Domestic Violence, 2021).  Victims can isolate, become depressed, or feel anxious around the abuser (National Network to End Domestic Violence, 2021). These manifestations of violence and abuse may be the red flags needed for nurses, health-care providers, and family members to recognize that abuse is occurring.
Effects of Violence on Mental Health
Psychologic abuse can lead to depression , anxiety , and suicidal ideation . People can feel confusion, shame, and guilt over being in the relationship (National Network to End Domestic Violence, 2021). The victim feels there is no way out of the horrible situation except death. Suicidal ideation can be a response to mental health conditions, life events, substance use problems, and violence. People at risk of suicide may be more likely to engage in risky behaviors due to their underlying mental health crisis. Nurses may recognize their patient is depressed, using drugs, and feels hopeless. These could be signs of abuse leading to thoughts of suicide. It is important to provide support and a safety plan for people who are at risk for suicide. Referrals to mental health providers are essential for these patients.
Effects of Violence on Cognition
Survivors of violence experience immediate consequences, but they also suffer long-term consequences. Violence can affect short-term memory, cognitive control, and emotion regulation by altering brain organization and communication between brain regions (Mattheiss et al., 2022). This leads to lower grades in school, poor performance at work, and financial issues. Survivors of violence may have difficulty in making decisions. In children, violence can impact the development of their mental functions and can increase the risk of developing personality disorders during adolescence (Zucchelli & Giuseppe, 2019).
Effects of IPV on Reproductive Health
Women who experience violence undergo stress. Stress induces the release of cortisol, which then causes inflammation. Recent research has shown a correlation between chronic stress and a dysregulation of endocrine-inflammatory-epigenetic changes, which may lead to the development and progression of endometriosis (Hutchinson et al., 2023). Infertility has been linked to IPV (Stellar et al., 2016). IPV has also been associated with unintended pregnancies, abortions, miscarriages, and STIs.
Infertility and IPV in Iran
Iran’s infertility rate is 24.9 percent, and its IPV frequency varies between 14 percent and 61 percent (Alijani et al., 2022). Alijani et al.’s 2022 study focused on women being treated at an infertility clinic at a university medical center. They were interviewed for IPV using a questionnaire-screening. According to this study, 88.9 percent of the infertile women reported physical violence, and 85.8 percent reported psychologic violence.
Effects of Violence on Families
People who experience IPV can struggle to be good parents due to their low self-esteem and lack of confidence. They often lack support because of the isolation that some perpetrators cause; therefore, when children are sick or mothers need help with small children, they have very few people to ask for help. Preschool children who observe domestic violence may experience nightmares, developmental delays, and loss of memory functioning (U.S. Department of Veterans Affairs, 2021). Children may have difficulty expressing their emotions, show signs of PTSD, and have a difficult time making friends (U.S. Department of Veterans Affairs, 2021). Adolescents may be forced into the role of adult and become the caretaker of younger children.  Nurses should consider resources for the whole family when discussing IPV with patients.
Centers for Disease Control and Prevention Interventions on Behalf of Women
The CDC has developed violence prevention programs to address IPV. Their site also includes facts about IPV, risks and protective factors, prevention strategies, discussions of different types of IPV, and specific programs for dating and teen violence (CDC, 2021a). The CDC’s prevention strategies include teaching safe and healthy relationship skills, engaging influential adults and peers, disrupting the developmental pathways toward partner violence, creating protective environments, strengthening economic supports for families, and supporting survivors to increase safety and lessen harm. The CDC (2021a) also developed the National Intimate Partner and Sexual Violence Survey (NISVS) to collect the most current and comprehensive data on IPV, sexual violence, and stalking in the United States. These data are collected to help create violence prevention strategies.
Violence Prevention Timeline
The CDC has developed a timeline devoted to mapping the progression of violence as a public health problem. The timeline begins in 1979 with the U.S. Surgeon General’s Report, Healthy People , which identified violence as one of 15 priority areas for improvement for the nation (CDC, 2022a). The CDC established the Division of Injury Epidemiology and Control to study injuries and violence in 1986. In 1994, Congress passed the Violence Against Women Act that aided in preventing IPV and provided grants for rape prevention and education (CDC, 2022a).
In 2000, the WHO also created a Department of Injuries and Violence Prevention. The timeline then presented the release of evidence showing the effects of IPV on public health and the programs that were developed in response to this research. The timeline continues to this day as the CDC continues to find ways to battle violence against women.
National Intimate Partner and Sexual Violence Survey
In 1998, a joint project between the National Institute of Justice and the CDC was created to discover more about IPV and sexual violence in the United States. The National Intimate Partner and Sexual Violence Survey was launched. The investigators used telephone surveys to learn more information about violence against women. The results showed the prevalence and incidence of rape, physical assault, and stalking; the rate of injury among those incidences; and injured victims’ use of health-care services (U.S. Department of Justice, 1998). The survey found that women from underrepresented groups had higher incidences of IPV and rape, that rape is more common for women under the age of 18, and that stalking is a significant social problem (U.S. Department of Justice, 1998). The survey provided data so that programs would be aimed at improving the negative data results.


20.3: Interstellar Gas
By the end of this section, you will be able to:
- Name the major types of interstellar gas
- Discuss how we can observe each type
- Describe the temperature and other major properties of each type
Interstellar gas, depending on where it is located, can be as cold as a few degrees above absolute zero or as hot as a million degrees or more. We will begin our voyage through the interstellar medium by exploring the different conditions under which we find gas.
Ionized Hydrogen (H II) Regions—Gas Near Hot Stars
Some of the most spectacular astronomical photographs show interstellar gas located near hot stars (Figure 20.3). The strongest line in the visible region of the hydrogen spectrum is the red line in the Balmer series 1 (as explained in the chapter on Radiation and Spectra); this emission line accounts for the characteristic red glow in images like Figure 20.3.
Hot stars are able to heat nearby gas to temperatures close to 10,000 K. The ultraviolet radiation from the stars also ionizes the hydrogen (remember that during ionization, the electron is stripped completely away from the proton). Such a detached proton won’t remain alone forever when attractive electrons are around; it will capture a free electron, becoming a neutral hydrogen once more. However, such a neutral atom can then absorb ultraviolet radiation again and start the cycle over. At a typical moment, most of the atoms near a hot star are in the ionized state.
Since hydrogen is the main constituent of interstellar gas, we often characterize a region of space according to whether its hydrogen is neutral or ionized. A cloud of ionized hydrogen is called an H II region . (Scientists who work with spectra use the Roman numeral I to indicate that an atom is neutral; successively higher Roman numerals are used for each higher stage of ionization. H II thus refers to hydrogen that has lost its one electron; Fe III is iron with two electrons missing.)
The electrons that are captured by the hydrogen nuclei cascade down through the various energy levels of the hydrogen atoms on their way to the lowest level, or ground state. During each transition downward, they give up energy in the form of light. The process of converting ultraviolet radiation into visible light is called fluorescence . Interstellar gas contains other elements besides hydrogen. Many of them are also ionized in the vicinity of hot stars; they then capture electrons and emit light, just as hydrogen does, allowing them to be observed by astronomers. But generally, the red hydrogen line is the strongest, and that is why H II regions look red.
A fluorescent light on Earth works using the same principles as a fluorescent H II region. When you turn on the current, electrons collide with atoms of mercury vapor in the tube. The mercury is excited to a high-energy state because of these collisions. When the electrons in the mercury atoms return to lower-energy levels, some of the energy they emit is in the form of ultraviolet photons. These, in turn, strike a phosphor-coated screen on the inner wall of the light tube. The atoms in the screen absorb the ultraviolet photons and emit visible light as they cascade downward among the energy levels. (The difference is that these atoms give off a wider range of light colors, which mix to give the characteristic white glow of fluorescent lights, whereas the hydrogen atoms in an H II region give off a more limited set of colors.)
Neutral Hydrogen Clouds
The very hot stars required to produce H II regions are rare, and only a small fraction of interstellar matter is close enough to such hot stars to be ionized by them. Most of the volume of the interstellar medium is filled with neutral (nonionized) hydrogen. How do we go about looking for it?
Unfortunately, neutral hydrogen atoms at temperatures typical of the gas in interstellar space neither emit nor absorb light in the visible part of the spectrum. Nor, for the most part, do the other trace elements that are mixed with the interstellar hydrogen. However, some of these other elements can absorb visible light even at typical interstellar temperatures. This means that when we observe a bright source such as a hot star or a galaxy, we can sometimes see additional lines in its spectrum produced when interstellar gas absorbs light at particular frequencies (see Figure 20.4). Some of the strongest interstellar absorption lines are produced by calcium and sodium, but many other elements can be detected as well in sufficiently sensitive observations (as discussed in Radiation and Spectra).
The first evidence for absorption by interstellar clouds came from the analysis of a spectroscopic binary star (see The Stars: A Celestial Census), published in 1904. While most of the lines in the spectrum of this binary shifted alternately from longer to shorter wavelengths and back again, as we would expect from the Doppler effect for stars in orbit around each other, a few lines in the spectrum remained fixed in wavelength. Since both stars are moving in a binary system, lines that showed no motion puzzled astronomers. The lines were also peculiar in that they were much, much narrower than the rest of the lines, indicating that the gas producing them was at a very low pressure. Subsequent work demonstrated that these lines were not formed in the star’s atmosphere at all, but rather in a cold cloud of gas located between Earth and the binary star.
While these and similar observations proved there was interstellar gas , they could not yet detect hydrogen, the most common element, due to its lack of spectral features in the visible part of the spectrum. (The Balmer line of hydrogen is in the visible range, but only excited hydrogen atoms produce it. In the cold interstellar medium, the hydrogen atoms are all in the ground state and no electrons are in the higher-energy levels required to produce either emission or absorption lines in the Balmer series.) Direct detection of hydrogen had to await the development of telescopes capable of seeing very-low-energy changes in hydrogen atoms in other parts of the spectrum. The first such observations were made using radio telescopes, and radio emission and absorption by interstellar hydrogen remains one of our main tools for studying the vast amounts of cold hydrogen in the universe to this day.
In 1944, while he was still a student, the Dutch astronomer Hendrik van de Hulst predicted that hydrogen would produce a strong line at a wavelength of 21 centimeters. That’s quite a long wavelength, implying that the wave has such a low frequency and low energy that it cannot come from electrons jumping between energy levels (as we discussed in Radiation and Spectra). Instead, energy is emitted when the electron does a flip, something like an acrobat in a circus flipping upright after standing on his head.
The flip works like this: a hydrogen atom consists of a proton and an electron bound together. Both the proton and the electron act is if they were spinning like tops, and spin axes of the two tops can either be pointed in the same direction (aligned) or in opposite directions (anti-aligned). If the proton and electron were spinning in opposite directions, the atom as a whole would have a very slightly lower energy than if the two spins were aligned (Figure 20.5). If an atom in the lower-energy state (spins opposed) acquired a small amount of energy, then the spins of the proton and electron could be aligned, leaving the atom in a slightly excited state . If the atom then lost that same amount of energy again, it would return to its ground state. The amount of energy involved corresponds to a wave with a wavelength of 21 centimeters; hence, it is known as the 21-centimeter line .
Neutral hydrogen atoms can acquire small amounts of energy through collisions with other hydrogen atoms or with free electrons. Such collisions are extremely rare in the sparse gases of interstellar space. An individual atom may wait centuries before such an encounter aligns the spins of its proton and electron. Nevertheless, over many millions of years, a significant fraction of the hydrogen atoms are excited by a collision. (Out there in cold space, that’s about as much excitement as an atom typically experiences.)
An excited atom can later lose its excess energy either by colliding with another particle or by giving off a radio wave with a wavelength of 21 centimeters. If there are no collisions, an excited hydrogen atom will wait an average of about 10 million years before emitting a photon and returning to its state of lowest energy. Even though the probability that any single atom will emit a photon is low, there are so many hydrogen atoms in a typical gas cloud that collectively they will produce an observable line at 21 centimeters.
Equipment sensitive enough to detect the 21-cm line of neutral hydrogen became available in 1951. Dutch astronomers had built an instrument to detect the 21-cm waves that they had predicted, but a fire destroyed it. As a result, two Harvard physicists, Harold Ewen and Edward Purcell , made the first detection (Figure 20.6), soon followed by confirmations from the Dutch and a group in Australia. Since the detection of the 21-cm line, many other radio lines produced by both atoms and molecules have been discovered (as we will discuss in a moment), and these have allowed astronomers to map out the neutral gas throughout our home Galaxy. Astronomers have also detected neutral interstellar gas, including hydrogen, at many other wavelengths from the infrared to the ultraviolet.
Modern radio observations show that most of the neutral hydrogen in our Galaxy is confined to an extremely flat layer, less than 300 light-years thick, that extends throughout the disk of the Milky Way Galaxy . This gas has densities ranging from about 0.1 to about 100 atoms per cm 3 , and it exists at a wide range of temperatures, from as low as about 100 K (–173 °C) to as high as about 8000 K. These regions of warm and cold gas are interspersed with each other, and the density and temperature at any particular point in space are constantly changing.
Ultra-Hot Interstellar Gas
While the temperatures of 10,000 K found in H II regions might seem warm, they are not the hottest phase of the interstellar medium. Some of the interstellar gas is at a temperature of a million degrees, even though there is no visible source of heat nearby. The discovery of this ultra-hot interstellar gas was a big surprise. Before the launch of astronomical observatories into space, which could see radiation in the ultraviolet and X-ray parts of the spectrum, astronomers assumed that most of the region between stars was filled with hydrogen at temperatures no warmer than those found in H II regions. But telescopes launched above Earth’s atmosphere obtained ultraviolet spectra that contained interstellar lines produced by oxygen atoms that have been ionized five times. To strip five electrons from their orbits around an oxygen nucleus requires a lot of energy. Subsequent observations with orbiting X-ray telescopes revealed that the Galaxy is filled with numerous bubbles of X-ray-emitting gas. To emit X-rays, and to contain oxygen atoms that have been ionized five times, gas must be heated to temperatures of a million degrees or more.
Theorists have now shown that the source of energy producing these remarkable temperatures is the explosion of massive stars at the ends of their lives (Figure 20.7). Such explosions, called supernovae , will be discussed in detail in the chapter on The Death of Stars. For now, we’ll just say that some stars, nearing the ends of their lives, become unstable and literally explode. These explosions launch gas into interstellar space at velocities of tens of thousands of kilometers per second (up to about 30% the speed of light). When this ejected gas collides with interstellar gas, it produces shocks that heat the gas to millions or tens of millions of degrees.
Astronomers estimate that one supernova explodes roughly every 100 years somewhere in the Galaxy. On average, shocks launched by supernovae sweep through any given point in the Galaxy about once every few million years. These shocks keep some interstellar space filled with gas at temperatures of millions of degrees, and they continually disturb the colder gas, keeping it in constant, turbulent motion.
Molecular Clouds
A few simple molecules out in space, such as CN and CH, were discovered decades ago because they produce absorption lines in the visible-light spectra of stars behind them. When more sophisticated equipment for obtaining spectra in radio and infrared wavelengths became available, astronomers—to their surprise—found much more complex molecules in interstellar clouds as well.
Just as atoms leave their “fingerprints” in the spectrum of visible light, so the vibration and rotation of atoms within molecules can leave spectral fingerprints in radio and infrared waves. If we spread out the radiation at such longer wavelengths, we can detect emission or absorption lines in the spectra that are characteristic of specific molecules. Over the years, experiments in our laboratories have shown us the exact wavelengths associated with changes in the rotation and vibration of many common molecules, giving us a template of possible lines against which we can now compare our observations of interstellar matter.
The discovery of complex molecules in space came as a surprise because most of interstellar space is filled with ultraviolet light from stars, and this light is capable of dissociating molecules (breaking them apart into individual atoms). In retrospect, however, the presence of molecules is not surprising. As we will discuss further in the next section, and have already seen above, interstellar space also contains significant amounts of dust capable of blocking out starlight. When this dust accumulates in a single location, the result is a dark cloud where ultraviolet starlight is blocked and molecules can survive. The largest of these structures are created where gravity pulls interstellar gas together to form giant molecular clouds , structures as massive as a million times the mass of the Sun. Within these, most of the interstellar hydrogen has formed the molecule H 2 (molecular hydrogen). Other, more complex molecules are also present in much smaller quantities.
Giant molecular clouds have densities of hundreds to thousands of atoms per cm 3 , much denser than interstellar space is on average. As a result, though they account for a very small fraction of the volume of interstellar space, they contain a significant fraction—20–30%—of the total mass of the Milky Way’s gas. Because of their high density, molecular clouds block ultraviolet starlight, the main agent for heating most interstellar gas. As a result, they tend to be extremely cold, with typical temperatures near 10 K (−263 °C). Giant molecular clouds are also the sites where new stars form, as we will discuss below.
It is in these dark regions of space, protected from starlight, that molecules can form. Chemical reactions occurring both in the gas and on the surface of dust grains lead to much more complex compounds, hundreds of which have been identified in interstellar space. Among the simplest of these are water (H 2 O), carbon monoxide (CO), which is produced by fires on Earth, and ammonia (NH 3 ), whose smell you recognize in strong home cleaning products. Carbon monoxide is particularly abundant in interstellar space and is the primary tool that astronomers use to study giant molecular clouds. Unfortunately, the most abundant molecule, H 2 , is particularly difficult to observe directly because in most giant molecular clouds, it is too cold to emit even at radio wavelengths. CO, which tends to be present wherever H 2 is found, is a much better emitter and is often used by astronomers to trace molecular hydrogen.
The more complex molecules astronomers have found are mostly combinations of hydrogen, oxygen, carbon, nitrogen, and sulfur atoms. Many of these molecules are organic (those that contain carbon and are associated with the carbon chemistry of life on Earth.) They include formaldehyde (used to preserve living tissues), alcohol (see the feature box on Cocktails in Space), and antifreeze.
In 1996, astronomers discovered acetic acid (the prime ingredient of vinegar) in a cloud lying in the direction of the constellation of Sagittarius. To balance the sour with the sweet, a simple sugar (glycolaldehyde) has also been found. The largest compounds yet discovered in interstellar space are fullerenes , molecules in which 60 or 70 carbon atoms are arranged in a cage-like configuration (see Figure 20.8). See Table 20.1 for a list of a few of the more interesting interstellar molecules that have been found so far.
| Some Interesting Interstellar Molecules | ||
|---|---|---|
| Name | Chemical Formula | Use on Earth |
| Ammonia | NH 3 | Household cleansers |
| Formaldehyde | H 2 CO | Embalming fluid |
| Acetylene | HC 2 H | Fuel for a welding torch |
| Acetic acid | C 2 H 2 O 4 | The essence of vinegar |
| Ethyl alcohol | CH 3 CH 2 OH | End-of-semester parties |
| Ethylene glycol | HOCH 2 CH 2 OH | Antifreeze ingredient |
| Benzene | C 6 H 6 | Carbon ring, ingredient in varnishes and dyes |
The cold interstellar clouds also contain cyanoacetylene (HC 3 N) and acetaldehyde (CH 3 CHO), generally regarded as starting points for amino acid formation. These are building blocks of proteins, which are among the fundamental chemicals from which living organisms on Earth are constructed. The presence of these organic molecules does not imply that life exists in space, but it does show that the chemical building blocks of life can form under a wide range of conditions in the universe. As we learn more about how complex molecules are produced in interstellar clouds, we gain an increased understanding of the kinds of processes that preceded the beginnings of life on Earth billions of years ago.
Interested in learning more about fullerenes, buckyballs, or buckminsterfullerenes (as they’re called)? Watch a brief video from NASA’s Jet Propulsion Laboratory that explains what they are and illustrates how they were discovered in space.
Among the molecules astronomers have identified in interstellar clouds is alcohol, which comes in two varieties: methyl (or wood) alcohol and ethyl alcohol (the kind you find in cocktails). Ethyl alcohol is a pretty complex molecule, written by chemists as C 2 H 5 OH. It is quite plentiful in space (relatively speaking). In clouds where it has been identified, we detect up to one molecule for every m 3 . The largest of the clouds (which can be several hundred light-years across) have enough ethyl alcohol to make 10 28 fifths of liquor.
We need not fear, however, that future interstellar astronauts will become interstellar alcoholics. Even if a spaceship were equipped with a giant funnel 1 kilometer across and could scoop it through such a cloud at the speed of light, it would take about a thousand years to gather up enough alcohol for one standard martini.
Furthermore, the very same clouds also contain water (H 2 O) molecules. Your scoop would gather them up as well, and there are a lot more of them because they are simpler and thus easier to form. For the fun of it, one astronomical paper actually calculated the proof of a typical cloud. Proof is the ratio of alcohol to water in a drink, where 0 proof means all water, 100 proof means half alcohol and half water, and 200 proof means all alcohol. The proof of the interstellar cloud was only 0.2, not enough to qualify as a stiff drink
Footnotes
- 1 Scientists also call this red Balmer line the H-alpha line, with alpha meaning it is the first spectral line in the Balmer series.


7.3: Order of Operations
Learning Outcomes
- Use the order of operations to correctly perform multi-step arithmetic.
- Apply the order of operations to statistics related complex questions.
When we are given multiple arithmetic operations within a calculation, there is an established order that we must do them in based on how the expression is written. Understanding these rules is especially important when using a calculator, since calculators are programmed to strictly follow the order of operations. This comes up in every topic in statistics, so knowing the order of operations is an essential skill for all successful statistics students to have.
PEMDAS
The order of operations are as follows:
- P arentheses
- E xponents
- M ultiplication and D ivision, from left to right
- A ddition and S ubtraction, from left to right
Notice that Multiplication and division are listed together as item 3. If you see multiplication and division in the same expression the rule is to go in order from left to right. You do not calculate all of the multiplication operations then go back and do the division. Similarly, if you see addition and subtraction in the same expression the rule is to go in order from go left to right. The same goes for two of the same arithmetic operators.
Example \(\PageIndex{1}\)
Evaluate: \(20-6\div3+\left(2\times3^2\right)\)
Solution
We start with what is inside the parentheses: \(2 \times 3^2\). Since exponents comes before addition, we find \(3^2=9\) first. We now have
\[20-6\div3+\left(2\times9\right) \nonumber\]
We continue inside the parentheses and perform the multiplication: \(2\times9=18\).
This gives
\[20-6\div3+18 \nonumber\]
Since division comes before addition and subtraction, we next calculate \(6\div3=2\) to get
\[20-2+18 \nonumber\]
Since subtraction and addition are tied, we go from left to right. We calculate: \(20-2=18\) to get
\[18+18\:=36 \nonumber\]
The key to arriving at the correct answer is to go slow and write down each step in the arithmetic.
Hidden Parentheses
You may think that since you always have a calculator or computer at hand, that you don't need to worry about order of operations. Unfortunately, the way that expressions are written is not the same as the way that they are entered into a computer or calculator. In particular, exponents need to be treated with care as do fractions bars.
Example \(\PageIndex{2}\)
Evaluate \(2.1^{6-2}\)
Solution
First, note that we use the symbol "^" to tell a computer or calculator to exponentiate. If you were to enter 2.1^6-2 into a computer, it would give you the answer of 83.766121 which is not correct, since the computer will first calculate \(2.1^6\) and then subtract 2. Since the subtraction is within the exponent, it must be performed first. To tell a calculator or computer to perform the subtraction first, we use parentheses:
2.1^(6 - 2) = 2.1 4 = 19.4481
Example \(\PageIndex{3}\)
The "z-score" is defined by:
\[z=\frac{x-\mu}{\sigma} \nonumber\]
Find the z-score rounded to one decimal place if:
\[x=2.323,\:\mu=1.297,\:\sigma=0.241 \nonumber\]
Solution
Once again, if we put these numbers into the z-score formula and use a computer or calculator by entering \(3.323\:-\:1.297\:\div\:0.241\) we will get -0.259 which is the wrong answer. Instead, we need to know that the fraction bar separates the numerator and the denominator, so the subtraction must be done first. We compute
\[\frac{2.323-1.297}{0.241}\:=\left(2.323-1.297\right)\div0.241=\:4.25726141 \nonumber\]
Now round to one decimal place to get 4.3. Notice that if you rounded before you did the arithmetic, you would get exactly 5 which is very different. 4.3 is more accurate.
Try It
Suppose the equation of the regression line for the number of pairs of socks a person owns, \(\widehat{y}\), based on the number of pairs of shoes, \(x\), the person owns is
\[\widehat{y}=6+2x \nonumber\]
Use this regression line equation to predict the number of pairs of socks a person owns for a person who owns 4 pairs of shoes.
- Answer
-
When \(x\) = 4, \(\widehat{y}=6+2(4) = 6 + 8 = 14\).
A person who owns 4 pairs of shoes has 14 pairs of socks, according to the regression model.
Additional video resources:


10.8: Review Questions
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- Last updated
- Save as PDF
1.
Carlos is thinking about a career in aviation. During the summer between his second and third years of high school, he completes an internship at an airport nearby. What identity status would you say he is working toward?
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foreclosure
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achievement
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diffusion/confusion
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moratorium
2 .
The theories described in the chapter see adolescence as a ________ .
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tumultuous but exciting time of life
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brief life stage
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life stage about which we know little
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time for cementing our political beliefs
3.
Fourteen-year-old Samantha is confident she will be a practicing psychotherapist one day. Which of Marcia’s proposed identity statuses is she in?
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moratorium
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foreclosure
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achievement
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diffusion/confusion
4 .
Which statement is most accurate with regard to rites of passage that are experienced during the adolescent years?
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They denote that a person has reached adulthood.
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They are found only in cultures from developed nations.
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There is considerable variation between different cultures.
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They are universal to virtually all cultures.
5.
Which statement is typical for an adolescent and not a child?
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I am fair, responsible, and trustworthy.
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I like trains, sports, and any type of fruit.
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I like to dance, swim, and play cards.
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I am a good person.
6 .
How does the level of self-esteem typically change across the adolescent period?
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low at first, quick rise, and then a leveling off near the end
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medium level, then a dip, then a rapid rise
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gradual increase with age
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decline from middle-childhood level
7.
Which best describes self-regulation in adolescents?
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The ability to act impulsively and follow immediate desires without considering consequences.
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The tendency to avoid all forms of external assistance in any situation, regardless of complexity.
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The capacity to rely entirely on parental guidance for behavioral control and decision-making.
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The skill to manage behavior and emotions to achieve better outcomes, influenced by brain developments and social factors.
8 .
When thirteen-year-old Taylor points out someone’s body size, their parent frequently responds, “Everyone is unique and beautiful in their own way!” What is Taylor’s parent acknowledging?
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self-love
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body diversity
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emotional intelligence
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somatic dimorphism
9.
 He wants to snap back at her, but he keeps quiet. “Talk to her after class and don’t make it worse,” he tells himself. After class he expresses his feelings firmly but respectfully, and the teacher apologizes profusely for having embarrassed him. Tar’darius feels better and goes about the rest of his day. This adolescent has shown a high level of ________.
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identity foreclosure
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self-esteem
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emotional intelligence
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working memory regulation
10 .
LGBTQ+ youth who are developing their sexual identity often accomplish relevant tasks at different points in time. What approach to identity development explains this?
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milestone approach
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stage approach
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life course approach
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differentiated aspect
11.
 She also wonders why her female classmates giggle at her when she sometimes spends time talking to and playing with boys in her class at recess. Which concept best explains such segregation?
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identity foreclosure
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gender conforming
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colorblind approach
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gender intensification hypothesis
12 .
For Black youth, entry into adolescence typically marks the beginning of the ________ phase of racial identity development, during which a negative trigger related to race is experienced.
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conflict
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encounter
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discovery
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rejection
13.
Code-switching, or changing certain aspects of one’s presentation or behavior, is often seen in people who are bi- or multi-racial. The research suggests that this approach to one’s various racial identities can be ________.
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denigrating
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counter-productive
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instinctive
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exhausting
14 .
What has the trend been between 2015 and 2021 in adolescent media use?
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increased screen time
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stable amount of screen time
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decreased screen time
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increased print material exposure
15.
Kevin texts his boyfriend at least once a day, and most of these texts are, “Do you still love me” and “Is everything okay between us.” What attachment style might Kevin be exhibiting?
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anxious-preoccupied
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avoidant
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disorganized
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secure
16 .
What is a significant risk factor for teen dating violence?
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high academic performance
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supportive and communicative family relationships
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prior experience with other forms of violence or abuse
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active participation in school extracurricular activities
17.
Suppose you noticed a table of a half-dozen or so football players in a school cafeteria. What is the term for this grouping of friends?
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clique
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crowd
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focus group
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social network
18 .
Which factor is correlated with absenteeism during high school?
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student’s height and weight
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externalizing and internalizing behaviors
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being an only child
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involvement in extracurricular activities
19.
What style of parenting is associated with the most positive outcomes for teens?
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authoritarian style
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authoritative style
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uninvolved style
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permissive style
20 .
What approach to understanding adolescent needs focuses on identifying their strengths?
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milestone approach
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developmental assets
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stage theory
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discontinuity model
21.
Marta has always wanted to follow in her mother’s footsteps and become a civil rights attorney, even though her mother never pushed this direction. She feels her mother does good for the world, and has a meaningful career. Now that she’s approaching the end of high school, her interest in that occupation has dwindled, and she thinks her mother must have a rather boring and monotonous job. This is an example of the ________ of parents that sometimes happens in adolescents.
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interpersonal strife
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uninvolvement
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de-idealization
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misprioritizing


3.2: Test-Taking Strategies
Learning Objectives
At the end of this section, the learner will:
- Review test-taking strategies for before, during, and after an exam.
- Identify strategies specific to different types of test items.
- Apply test-taking tips and strategies to a scenario.
Test Taking Strategies
Approaching a test in a careful and methodical way can help ensure the best results. You can best utilize your exam time by taking the time to survey the exam directions and point values and planning how to approach sections of the test. Strategy is particularly important for tests with mixed types of questions (i.e., multiple choice and essay) or tests with multiple essay questions. For example, if you were to spend too much time on the multiple choice questions and not allocate enough time to complete an essay question, you may have answered the majority of the total number of questions but still do poorly on the exam if the essay question was worth a larger portion of the exam’s total points. Plan your time carefully and manage your stress.
Additionally, a lot of students are unaware of the many strategies available to help with the test-taking experience before, during, and after.
Effective Techniques for Test Preparation and Taking
Before the Test:
- Go to all of your classes and take lecture notes and read all of your assigned reading using techniques described in Chapters 3.2 and 3.3 .
- Develop a study plan and schedule the time to review for an exam over several days. Refer to “Create a Five-Day Study Plan for Exams in Chapter 3.4 .
- Schedule study sessions with classmates in advance of the night before the exam.
- If you know how many questions, what the format is, and/or how much time you will have, you can start to mentally prepare for the exam much more so than if you are coming in with no information. There are two more important aspects that you may or may not know: a) what will be covered or asked on the exam; b) how the exam will be scored.
- Be sure you have materials you need for the exam ready at least the night before the exam, such as multiple writing utensils, scantrons, blue books, notebook paper, etc.
- Get a good night’s rest the night before the exam.
- To ensure that you are on time to your exam, set two alarm clocks or arrange to have a friend call you to make sure you are awake.
- Eat a good breakfast and avoid caffeine prior to the exam.
- Prioritize self-care and test-anxiety strategies reviewed in previous chapters.
During the Test:
- Take a moment to take some deep cleansing breaths (or any other quick relaxation techniques) if you feel stressed by a question or the exam as a whole.
- Keep your eye on the clock. Sit where you are most comfortable and have less distractions.
- Wear earplugs, if noise distracts you. If permitted, get up and stretch (or stretch in your chair) from time to time to relieve tension and assist the blood to the brain!
- Always read the directions first. Mark your exam with the necessary identification. Read them thoroughly. Scan the exam for question types, point values, etc.
- Develop a plan for how you will use your time to complete all questions on the exam.
- Read each item carefully and fully before marking an answer. You might find clues to the answer and will be less likely to make a preventable mistake.
- Skip difficult questions and come back to them if you are pressed for time. Caution: if using a scantron to record answers be very careful that you do not incorrectly record your answers on the wrong line and get off track.
- If time is available at the end of the exam, review questions you marked as “difficult” in order to check for errors. Try not to leave any answers blank. You might be awarded partial credit or even make a correct guess.
- If you have an essay question to write, take the time to briefly outline an answer to ensure that you are answering all aspects of the question and that you provide a well-organized answer for your faculty member to read.
After the Test:
- Conduct an error analysis for items missed on the test. Look up the correct answers. Determine the nature of the errors you made.
- Talk with your professor about items you missed on the exam. Discuss adjustments in your test preparation methods if necessary. This is especially useful for quizzes that contain information that may be incorporated into more inclusive exams such as mid-terms and finals.
- Analyze the test questions to see if patterns arise, such as, “Were most of the questions from the lecture, the book, or both?” “What types of questions were asked?” “What levels of thinking were being tested based on Bloom’s Taxonomy?” etc.
- Review the methods that you used to study for the exam and determine what adjustments you need to make in preparing for the next exam.
- Determine your overall grade in the course based on the new information that this grade has provided you. Talk with your professor and academic advisor if you have concerns about your overall grade.
- If allowed, archive your exam for future reference. Students should review the answers that were correct because they may see those questions on future exams and it is important to reinforce learning . Students should also review the answers that were incorrect in order to learn what the correct answer was and why.
Tips for Better Test TakinG: types of test items
The types of test items on a test are as different as the professors who created the tests. “Objective” test questions refer to items in which students have to recognize the correct answer from a list of provided options (e.g., multiple choice, true-false, matching). In most cases there is one best answer, though it is important to note that a professor could indicate in the directions that you can select more than one answer. “Subjective” test items are the fill-in-the-blank, short answer, or essay questions in which students must recall and produce the answer.
Multiple Choice Questions
- Read the question and all of the answer options first.
- Think of these questions as four true or false statements in one. One of the statements is true (the correct answer) and the others will be false. Mark out any answers that you know are not correct.
- Once you have selected an answer, do not change it unless you misread the question and know the new answer is correct. Usually, your first answer is the correct one if you are making an educated guess.
- Many times, the correct answer has more information in it.
- If there is no penalty for guessing, do not leave any items blank—make an educated guess.
- If there is an “All of the Above” option and there are two options that are correct, select “All of the Above.”
- In a question with “All of the Above” and “None of the Above,” if you are certain that one of the answers is true, do not select “None of the Above” and, likewise, if one of the answers is false, then do not select “All of the Above.”
- Make sure to match the grammar of question and answer. For example, if the question indicates a plural answer, look for the plural answer.
True-False Questions
- Read the statement carefully, but do not read too much into the statement. Base your answer on the information provided.
- Make sure to read the entire statement. With statements that have multiple facts, all parts of a sentence must be true if the whole statement is to be true. If one part of it is false, the whole sentence is false. Long sentences are often false for this reason.
- Qualifying words like “all”, “always,” “never,” “no”, “none”, “only”, and “every” indicate that this would have to be true all of the time. If it is not true all of the time, then you should answer false.
- Qualifying words like “usually,” “sometimes,” “many”, “most”, “some”, “often”, and “generally” indicate that it could be true or false depending on the situation. Oftentimes the answer is true.
Matching Questions
- First, read the instructions and take a look at both lists to determine what the items are and their relationship. It is especially important to determine if both lists have the same number of items and if all items are to be used, and used only once.
- Count both sides to be matched. Matching exams become much more difficult if one list has more items than the other or if items either might not be used or could be used more than once. If your exam instructions do not address this, you may wish to ask your instructor for further clarification.
- Take a look at the whole list before selecting an answer because a more correct answer may be found further into the list.
- Mark items when you are sure you have a match (pending the number of items in the list this may eliminate answers for the future). Guessing (if needed) should take place once you have selected answers you are certain about.
Essay Questions
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- Read the essay question(s) and the instructions first. Make sure you understand what the question is asking you to do. An essay question is not an opportunity to simply regurgitate everything you know about a particular topic.
- Take a minute or so to plan out your answer and jot down a brief outline of key points to guide your writing before you start. A well-organized answer tends to score more points when graded.
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Formula for Essay Questions:
- Introductory statements that define terms and describe what you will accomplish in your essay. This section should be brief. Address the answer to the question in your first or second sentence.
- State your first main idea, and then give supporting facts, examples, statistics, or details. Follow with your next main idea and continue the pattern until complete. The bulk of your time should be spent developing this section of the essay as it would likely include the details your professor wants to see in grading. Typically, each main idea would have its own paragraph. Use transitional works like “first, second, third…,” “next,” “also,” “however,” etc., to aid your reader.
- End with a summary or final conclusion. This section is also likely to be brief.
- Budget time to proofread and revise if needed.
- Write legibly. Neater papers tend to receive higher marks.
- Save some time for review when you have finished writing to check spelling, grammar and coherent thought in your answer.
- Make sure you have addressed all parts of the essay question.
Activity 4.2 – Test-Taking Tips
Test-Taking Tips
Write an email with advice to this friend, offering test-taking tips and strategies you think will help him.
RJ believes he is good at organization, and he usually is–for about the first two weeks of classes. He then becomes overwhelmed with all of the handouts and materials and tends to start slipping in the organization department. When it comes to tests, he worries that his notes might not cover all of the right topics and that he will not be able to remember all of the key terms and points–especially for his math class. During tests, he sometimes gets stuck on an item and tends to spend too much time there. He also sometimes changes answers but finds out later that his original selection was correct. He is also easily distracted by other students and noises which makes it hard for him to concentrate sometimes, and, unfortunately, he does admit to occasionally “cramming” the night before.


2.6: Commutative, associative and distributive laws
In this short section we re-emphasise the shift away from blind calculation, and towards consideration of the structure of arithmetic, which was already implicit in Problems 7 — 10 , and Problems 16 — 17 in Chapter 1 .
Problem 60 Each of two positive numbers a and b is increased by 10%.
(i) What is the change of their sum ?
(ii) What is the percentage change of their product ?
(iii) What is the percentage change in their quotient ?
Problem 61 The numbers a, b, c, d, e, f are positive. How will the value of the expression
change if the value of f is doubled?
Problem 62 In Problem 17 we saw that it is no accident that the sum of entries in the 4 by 4 ‘multiplication table’ is equal to 100.
(a) Go back to the proof that the total is equal to and see how this depends on the distributive law.
(b) The total of all entries in the multiplication square can be broken down into a succession of “reverse L-shapes”, such as the one formed by the bottom row and right hand column (shown above in bold ).
(i) Work out the subtotal in each of the four reverse L-shapes in the 4 by 4 multiplication table. What do you notice about these four subtotals?
(ii) Use the formulae for the k th and ( k — 1) th triangular numbers T k and T k - 1 to prove that, in the n by n multiplication table, the k th reverse L-shape always gives rise to a subtotal k 3 .
Conclude that
Hence find a simple formula for the sum C n of the first n cubes.
Now that we have a compact formula
- for the sum T n of the first n positive integers, and
- for the sum C n of the first n positive cubes,
we would naturally like to find a similar formula
- (that is, the sum of the entries on the leading diagonal of the n by n multiplication square).
This can be surprisingly elusive. But one way of obtaining it is to look instead for the sum of the entries in the sloping diagonal 2, 6, 12, 20, ... just above the main diagonal in the n by n multiplication square.
Problem 63 Consider the n by n multiplication square.
(a) Express the r th term in the sloping diagonal just above the main diagonal in terms of r . Hence show that the sum of entries in this sloping diagonal is equal to .
(b) Multiply by 3 each of the terms in the sloping diagonal just above the main diagonal.
(i) Guess a formula for the successive sums of these terms (6, 6 + 18, 6 + 18 + 36, ...), and prove that your formula is correct.
(ii) Hence derive a formula for the sum of the first n squares.


1.16: Deleting Files and Folders
Learning Outcomes
- Delete and restore files and folders using the Recycle Bin.
To delete a file or folder (or multiple selected files), right-click on the file and select Delete. You can also select the file and hit the Delete key on the keyboard. Deleting a folder deletes all its contents as well.
You may get a dialog prompt that asks if you want to move the file to the recycling bin. If that happens, say yes. If you didn’t get a dialog prompt, the file was still sent to the Recycle Bin. (The Mac equivalent is called Trash.) Deleted files are not gone forever ; they are sent to the Recycle Bin.
Recycling Bin
Deleted files are not gone forever. When a file is deleted, it is sent to the Recycle Bin. The Recycle Bin can be found on the Desktop. Its icon resembles a recycle bin or trash can, and the icon changes depending on whether the Recycle Bin contains any files.
The recycle bin is a holding place for deleted files. To permanently delete a file, you can delete it again from the Recycle Bin, or you can empty the Recycle Bin. It is recommended you empty the Recycle Bin every once in a while to free up space.
If you delete something by accident, or if you change your mind about a deleted file, you can un-delete it by restoring the file from the Recycle Bin.
To restore or permanently delete a file, right-click on the file and select either Restore or Delete. Deleted files are permanently deleted; restored files are restored to their original location before being deleted.
You can restore or permanently delete individual files or multiple files at once using the Manage option on the toolbar.
In the toolbar in the screenshot above, from left to right:
- Empty Recycle Bin: Permanently deletes everything in the Recycle Bin.
- Recycle Bin properties: Allows you to choose how much space to allot to the Recycle Bin. When the Recycle Bin reaches capacity, it will automatically delete its oldest files. The Recycle Bin properties also has an option to automatically delete files permanently. (Unless you know what you are doing, it is recommended you leave Recycle Bin properties alone.)
- Restore all items: Restores everything in the Recycle Bin to its former location.
- Restore the selected items: Restores only the file or files you have selected.
Contributors and Attributions
- Deleting Files. Authored by : Lumen Learning. License : CC BY: Attribution


13.2: Inradius of h-triangle
The inradius of any h-triangle is less than \(\dfrac{1}{2} \cdot \ln3\) .
- Proof
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Let \(I\) and \(r\) be the h-incenter and h-inradius of \(\triangle_hXYZ\) .
Note that the h-angles \(XIY\) , \(YIZ\) and \(ZIX\) have the same sign. Without loss of generality, we can assume that all of them are positive and therefore
\(\measuredangle_hXIY+ \measuredangle_hYIZ+ \measuredangle_hZIX=2 \cdot \pi\)
We can assume that \(\measuredangle_hXIY \ge \dfrac{2}{3} \cdot \pi\) ; if not relabel \(X\) , \(Y\) , and \(Z\) .
Since \(r\) is the h-distance from \(I\) to \((XY)_h\) , Proposition 13.1.1 implies that
\(\begin{array} {rcl} {r} & < & {\dfrac{1}{2} \cdot \ln \dfrac{1 + \cos \dfrac{\pi}{3}}{1 - \cos \dfrac{\pi}{3}}} \\ {} & = & {\dfrac{1}{2} \cdot \ln \dfrac{1 + \dfrac{1}{2}}{1 - \dfrac{1}{2}}} \\ {} & = & {\dfrac{1}{2} \cdot \ln 3.} \end{array}\)
Let \(\square_h ABCD\) be a quadrangle in the h-plane such that the h-angles at \(A\) , \(B\) , and \(C\) are right and \(AB_h=BC_h\) . Find the optimal upper bound for \(AB_h\) .
- Hint
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Note that the angle of prarllelism of \(B\) to \((CD)_h\) is bigger than \(\dfrac{\pi}{4}\), and it converge to \(\dfrac{\pi}{4}\) as \(CD_h \to \infty\).
Applying Proposition 13.1.1 , we get that
\(BC_h < \dfrac{1}{2} \cdot \ln \dfrac{1 + \dfrac{1}{\sqrt{2}}}{1 - \dfrac{1}{\sqrt{2}}} = \ln (1 + \sqrt{2}).\)
The right hand side is the limit of \(BC_h\) if \(CD_h \to \infty\). Therefore, \(\ln (1 + \sqrt{2})\) is the optimal upper bound.


2.7: Sorting News From Opinion
Traditional news organizations publish different kinds of articles in sections that have different expectations for writers and different editors selecting and shaping articles. Newsroom staff gather and report news (and newsrooms also publish stories by “stringers” or freelance journalists who are not on staff but are real, live journalists committed to the same values and practices as staffers). Some newsroom journalists don’t focus on “hard news” – they write features about style or business or the arts or sports. They also may publish analysis or in-depth investigative articles about things that aren’t breaking news of the day but help readers understand the complexities of newsworthy issues. A reporter may spend months looking into why kids in a particular neighborhood have high rates of asthma. Another may analyze why voters found a particular candidate appealing or why a policy decision in the US Congress is being held up. Those articles are not straight news reporting, but they aren’t opinion, either. Sources are provided, facts are checked. In fact, this example of fact-checking an investigative piece shows just how much work is involved to get the story right.
Source: Louise Kiernan, “ Calculating the Work Behind Our Work ,” ProPublica Illinois, December 29, 2017.
News stories usually don’t get as extensive a treatment; there’s just not enough time to retrace every step before the story goes out. However, editors will look for weaknesses and demand key facts be checked if they seem wobbly. There should be confirmation of any factual claim a source has made. Nothing should be taken at face value. There’s an old saying among journalists: if your mother says she loves you, check it out.
Opinion pieces and editorials (op/ed for short) are different. An editorial, strictly speaking, expresses the opinion of an editorial board. The board of the Wall Street Journal may write an article stating “we should do this” but their reporters would get in trouble if they tried to sneak that opinion into their reporting. An opinion piece is written by someone who presumably has both expertise and a point they want to make.
Opinion pieces make arguments. They recommend or persuade or simply make people think in a different way. Some opinion pieces are written by writers who are hired to share their thoughts on a regular basis – columnists like Tom Friedman or Michelle Alexander who write for The New York Times . Others are written by guests to the publication. A good news organization will not publish an argument that relies on lies or has no evidence for claims, but they go out of their way to publish arguments that their editorial board disagrees with so readers will encounter a range of approaches to ideas.
That’s how traditional news organizations make distinctions. There are also mixed sources such as The New Yorker which publishes different types of content, including fiction, reviews, essays, and satire, as well as thoroughly-sourced news stories and analysis. New media publications that have a less formal presentation – Quartz, Vice, Buzzfeed and the like – blur these distinctions more than traditional news organizations do. It’s harder to tell what’s news and what’s opinion, though sometimes these upstarts break a major news story. Likewise, on cable news it can be hard to distinguish a statement by a talk show host or an assertion by a pundit brought in for commentary from a statement made by a reporter. But they are distinctions that matter. Ask yourself: Who said that? What job were they doing when they said it?
When thinking about the information we get through networks, it’s worthwhile thinking about what kind of information it is. Is it news reporting? Is it opinion? Is it an advertisement dressed up to fool you into thinking it’s news? Is it disinformation camouflaged as news? Always take a moment to know what type of information you’re dealing with before you run with it.
When in doubt about a particular news source, see if Wikipedia has an article about it that describes its focus and reputation. If you want to see behind the scenes, click on the Talk tab to see if Wikipedians have argued about how to describe that source.