11.E: Sequences, Probability and Counting Theory (Exercises)
11.1 Sequences and Their Notations
Verbal
1) Discuss the meaning of a sequence. If a finite sequence is defined by a formula, what is its domain? What about an infinite sequence?
- Answer
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A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers.
2) Describe three ways that a sequence can be defined.
3) Is the ordered set of even numbers an infinite sequence? What about the ordered set of odd numbers? Explain why or why not.
- Answer
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Yes, both sets go on indefinitely, so they are both infinite sequences.
4) What happens to the terms \(a_n\) of a sequence when there is a negative factor in the formula that is raised to a power that includes \(n\)?
5) What is a factorial, and how is it denoted? Use an example to illustrate how factorial notation can be beneficial.
- Answer
-
A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out \(13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\)
Algebraic
For the exercises 6-15, write the first four terms of the sequence.
6) \(a_n=2^n-2\)
7) \(a_n=-\dfrac{16}{n+1}\)
- Answer
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First four terms: \(-8\), \(−\dfrac{16}{3}\), \(−4\), \(−\dfrac{16}{5}\)
8) \(a_n=-(-5)^{n-1}\)
9) \(a_n=\dfrac{2^n}{n^3}\)
- Answer
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First four terms: \(2\), \(\dfrac{1}{2}\), \(\dfrac{8}{27}\), \(\dfrac{1}{4}\)
10) \(a_n=\dfrac{2n+1}{n^3}\)
11) \(a_n=1.25\cdot (-4)^{n-1}\)
- Answer
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First four terms: \(1.25\), \(-5\), \(20\), \(-80\)
12) \(a_n=-4\cdot (-6)^{n-1}\)
13) \(a_n=\dfrac{n^2}{2n+1}\)
- Answer
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First four terms: \(\dfrac{1}{3}\), \(\dfrac{4}{5}\), \(\dfrac{9}{7}\), \(\dfrac{16}{9}\)
14) \(a_n=(-10)^n+1\)
15) \(a_n=-\left ( \dfrac{4\cdot (-5)^{n-1}}{5} \right )\)
- Answer
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First four terms: \(-\dfrac{4}{5}\), \(4\), \(-20\), \(100\)
For the exercises 16-20, write the first eight terms of the piecewise sequence.
16) \(a_n=\begin{cases} (-2)^n-2 & \text{ if } n \text{ is even} \\ (3)^{n-1} & \text{ if } n \text{ is odd} \end{cases}\)
17) \(a_n=\begin{cases} \dfrac{n^2}{2n+1} & \text{ if } n\leq 5 \\ n^2-5 & \text{ if } n>5 \end{cases}\)
- Answer
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\(\dfrac{1}{3}, \dfrac{4}{5}, \dfrac{9}{7}, \dfrac{16}{9}, \dfrac{25}{11}, 31, 44, 59\)
18) \(a_n=\begin{cases} (2n+1)^2 & \text{ if } n \text{ is divisible by } 4 \\ \dfrac{2}{n} & \text{ if } n \text{ is not divisible by } 4 \end{cases}\)
19) \(a_n=\begin{cases} -0.6\cdot 5^{n-1} & \text{ if } n \text{ is prime or } 1 \\ 2.5\cdot (-2)^{n-1} & \text{ if } n \text{ is composite } \end{cases}\)
- Answer
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\(−0.6,−3,−15,−20,−375,−80,−9375,−320\)
20) \(a_n=\begin{cases} 4(n^2-2) & \text{ if } n\leq 3 \text{ or } n>6 \\ \dfrac{n^2-2}{4} & \text{ if } 3<n\leq 6 \end{cases}\)
For the exercises 21-25, write an explicit formula for each sequence.
21) \(4, 7, 12, 19, 28,\ldots\)
- Answer
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\(a_n = n^2 + 3\)
22) \(-4,2,-10,14,-34,\ldots\)
23) \(1,1,\dfrac{4}{3},2,\dfrac{16}{5},\ldots\)
- Answer
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\(a_n=\dfrac{2^n}{2n} \text{ or } \dfrac{2^{n-1}}{n}\)
24) \(0,\dfrac{1-e^1}{1+e^2}, \dfrac{1-e^2}{1+e^3}, \dfrac{1-e^3}{1+e^4}, \dfrac{1-e^4}{1+e^5},\ldots\)
25) \(1,-\dfrac{1}{2}, \dfrac{1}{4}, -\dfrac{1}{8}, \dfrac{1}{16},\ldots\)
- Answer
-
\(a_n=\left ( -\dfrac{1}{2} \right )^{n-1}\)
For the exercises 26-30, write the first five terms of the sequence.
26) \(a_1=9, a_n=a_{n-1}+n\)
27) \(a_1=3, a_n=(-3)a_{n-1}\)
- Answer
-
First five terms: \(3, -9, 27, -81, 243\)
28) \(a_1=-4, a_n=\dfrac{a_{n-1}+2n}{a_{n-1}-1}\)
29) \(a_1=-1, a_n=\dfrac{(-3)^{n-1}}{a_{n-1}-2}\)
- Answer
-
First five terms: \(-1, 1, -9,\dfrac{27}{11},\dfrac{891}{5}\)
30) \(a_1=-30, a_n=(2+a_{n-1})\left (\dfrac{1}{2} \right )^n\)
For the exercises 31-33, write the first eight terms of the sequence.
31) \(a_1=\dfrac{1}{24},a_2=1, a_n=(2a_{n-2})(3a_{n-1})\)
- Answer
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\(\dfrac{1}{24},1,\dfrac{1}{4},\dfrac{3}{2},\dfrac{9}{4},\dfrac{81}{4},\dfrac{2187}{8},\dfrac{531,441}{16}\)
32) \(a_1=-1,a_2=5, a_n=a_{n-2}(3-a_{n-1})\)
33) \(a_1=2,a_2=10, a_n=\frac{2(a_{n-1}+2)}{a_{n-2}}\)
- Answer
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\(2,10,12,\dfrac{14}{5},\dfrac{4}{5},2,10,12\)
For the exercises 34-38, write a recursive formula for each sequence.
34) \(-2.5,-5,-10,-20,-40,\ldots\)
35) \(-8,-6,-3,1,6,\ldots\)
- Answer
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\(a_1=-8, a_n=a_{n-1}+n\)
36) \(2,4,12,48,240,\ldots\)
37) \(35,38,41,44,47,\ldots\)
- Answer
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\(a_1=35, a_n=a_{n-1}+3\)
38) \(15,3,\dfrac{3}{5},\dfrac{3}{25},\dfrac{3}{125},\ldots\)
For the exercises 39-42, evaluate the factorial.
39) \(6!\)
- Answer
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\(720\)
40) \(\left ( \dfrac{12}{6} \right )!\)
41) \(\dfrac{12!}{6!}\)
- Answer
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\(665,280\)
42) \(\dfrac{100!}{99!}\)
For the exercises 43-46, write the first four terms of the sequence.
43) \(a_n=\dfrac{n!}{n^2}\)
- Answer
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First four terms: 1, \(\dfrac{1}{2}\), \(\dfrac{2}{3}\), \(\dfrac{3}{2}\)
44) \(a_n=\dfrac{3\cdot n!}{4\cdot n!}\)
45) \(a_n=\dfrac{n!}{n^2 - n - 1}\)
- Answer
-
First four terms: -1, 2, \(\dfrac{6}{5}\), \(\dfrac{24}{11}\)
46) \(a_n=\dfrac{100\cdot n}{n(n-1)!}\)
Graphical
For the exercises 47-51, graph the first five terms of the indicated sequence
47) \(a_n=\dfrac{(-1)^n}{n}+n\)
- Answer
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48) \(a_n=\begin{cases} \dfrac{4+n}{2n} & \text{ if } n \text{ is even } \\ 3+n & \text{ if } \text{ if } n \text{ is odd } \end{cases}\)
49) \(a_1 = 2, a_n = (-a_{n-1} + 1)^2\)
- Answer
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50) \(a_n = 1, a_n = a_{n-1} + 8\)
51) \(a_n=\dfrac{(n+1)!}{(n-1)!}\)
- Answer
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For the exercises 52-54, write an explicit formula for the sequence using the first five points shown on the graph.
52)
53)
- Answer
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\(a_n=2^{n-2}\)
54)
For the exercises 55-56, write a recursive formula for the sequence using the first five points shown on the graph.
55)
- Answer
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\(a_1=6, a_n=2a_{n-1}-5\)
56)
Technology
Follow these steps to evaluate a sequence defined recursively using a graphing calculator:
- On the home screen, key in the value for the initial term \(a_1\) and press [ENTER] .
- Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes [2ND] ANS for the previous term \(a_{n-1}\). Press [ENTER] .
- Continue pressing [ENTER] to calculate the values for each successive term.
For the exercises 57-61, use the steps above to find the indicated term or terms for the sequence.
57) Find the first five terms of the sequence \(a_1=\dfrac{87}{111}\), \(a_n=\dfrac{4}{3}a_{n-1}+\dfrac{12}{37}\). Use the > Frac feature to give fractional results.
- Answer
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First five terms: \(\dfrac{29}{37},\dfrac{152}{111},\dfrac{716}{333},\dfrac{3188}{999},\dfrac{13724}{2997}\)
58) Find the \(15^{th}\) term of the sequence \(a_1=625, a_n=0.8a_{n-1}+18\).
59) Find the first five terms of the sequence \(a_1=2, a_n=2^{[(a_n-1)-1]}+1\).
- Answer
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First five terms: \(2,3,5,17,65537\)
60) Find the first ten terms of the sequence \(a_1=8, a_n=\frac{(a_{n-1}+1)!}{a_{n-1}!}\).
61) Find the tenth term of the sequence \(a_1=2, a_n=na_{n-1}\).
- Answer
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\(a_{10}=7,257,600\)
Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.
- In the home screen, press [2ND] LIST .
- Scroll over to OPS and choose “ seq( ” from the dropdown list. Press [ENTER] .
- In the line headed “ Expr: ” type in the explicit formula, using the \([X,T,\theta ,n]\) button for \(n\)
- In the line headed “ Variable: ” type in the variable used on the previous step.
-
In the line headed “
start:
” key in the value of n
that begins the sequence.n n -
In the line headed “
end:
” key in the value of n
that ends the sequence.n n - Press [ENTER] \(3\) times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms.
Using a TI-83, do the following.
- In the home screen, press [2ND] LIST .
- Scroll over to OPS and choose “ seq( ” from the dropdown list. Press [ENTER] .
- Enter the items in the order “ Expr ”, “ Variable ”, “ start ”, “end” separated by commas. See the instructions above for the description of each item.
- Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms.
For the exercises 62-66, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.
62) List the first five terms of the sequence \(a_n=-\dfrac{28}{9}n+\dfrac{5}{3}\).
63) List the first six terms of the sequence \(a_n=\dfrac{n^3-3.5n^2+4.1n-1.5}{2.4n}\).
- Answer
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First six terms: \(0.042,0.146,0.875,2.385,4.708\)
64) List the first five terms of the sequence \(a_n=\dfrac{15n\cdot (-2)^{n-1}}{47}\).
65) List the first four terms of the sequence \(a_n=5.7^n+0.275(n-1)!\)
- Answer
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First four terms: \(5.975,32.765,185.743,1057.25,6023.521\)
66) List the first six terms of the sequence \(a_n=\dfrac{n!}{n}\).
Extensions
67) Consider the sequence defined by \(a_n=-6-8n\). Is \(a_n=-421\) a term in the sequence? Verify the result.
- Answer
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If \(a_n=-421\) is a term in the sequence, then solving the equation \(-421=-6-8n\) for \(n\) will yield a non-negative integer. However, if \(-421=-6-8n\),
then \(n=51.875\) so \(a_n=-421\) is not a term in the sequence.
68) What term in the sequence \(a_n=\dfrac{n^2+4n+4}{2(n+2)}\) has the value \(41\)? Verify the result.
69) Find a recursive formula for the sequence \(1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1,\ldots\) ( Hint: find a pattern for \(a_n\) based on the first two terms.)
- Answer
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\(a_1=1, a_2=0, a_n=a_{n-1}-a_{n-2}\)
70) Calculate the first eight terms of the sequences \(a_n=\dfrac{(n+2)!}{(n-1)!}\) and \(b_n=n^3+3n^2+2n\), and then make a conjecture about the relationship between these two sequences.
71) Prove the conjecture made in the preceding exercise.
- Answer
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\(\dfrac{(n+2)!}{(n-1)!}=\dfrac{(n+2)\cdot (n+1)\cdot (n)\cdot (n-1)\cdot \ldots 3\cdot 2\cdot 1}{(n-1)\cdot \ldots 3\cdot 2\cdot 1}=n(n+1)(n+2)=n^3+3n^2+2n\)
11.2 Arithmetic Sequences
Verbal
1) What is an arithmetic sequence?
- Answer
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A sequence where each successive term of the sequence increases (or decreases) by a constant value.
2) How is the common difference of an arithmetic sequence found?
3) How do we determine whether a sequence is arithmetic?
- Answer
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We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.
4) What are the main differences between using a recursive formula and using an explicit formula to describe an arithmetic sequence?
5) Describe how linear functions and arithmetic sequences are similar. How are they different?
- Answer
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Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.
Algebraic
For the exercises 6-7, find the common difference for the arithmetic sequence provided.
6) \(\left \{ 5,11,17,23,29,... \right \}\)
7) \(\left \{ 0,\dfrac{1}{2},1,\frac{3}{2},2,... \right \}\)
- Answer
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The common difference is \(\dfrac{1}{2}\)
For the exercises 8-9, determine whether the sequence is arithmetic. If so find the common difference.
8) \(\left \{ 11.4,9.3,7.2,5.1,3,... \right \}\)
9) \(\left \{ 4,16,64,256,1024,... \right \}\)
- Answer
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The sequence is not arithmetic because \(16-4\neq 64-16\)
For the exercises 10-11, write the first five terms of the arithmetic sequence given the first term and common difference.
10) \(a_1=-25,d=-9\)
11) \(a_1=0,d=\dfrac{2}{3}\)
- Answer
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\(0, \dfrac{2}{3}, \dfrac{4}{3}, 2, \dfrac{8}{3}\)
For the exercises 12-13, write the first five terms of the arithmetic series given two terms.
12) \(a_1=17, a_7=-31\)
13) \(a_{13}=-60, a_{33}=-160\)
- Answer
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\(0,-5,-10,-15,-20\)
For the exercises 14-18, find the specified term for the arithmetic sequence given the first term and common difference.
14) First term is \(3\), common difference is \(4\), find the \(5^{th}\) term.
15) First term is \(4\), common difference is \(5\), find the \(4^{th}\) term.
- Answer
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\(a_4=19\)
16) First term is \(5\), common difference is \(6\), find the \(8^{th}\) term.
17) First term is \(6\), common difference is \(7\), find the \(6^{th}\) term.
- Answer
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\(a_6=41\)
18) First term is \(7\), common difference is \(8\), find the \(7^{th}\) term.
For the exercises 19-23, find the first term given two terms from an arithmetic sequence.
19) Find the first term or \(a_1\) of an arithmetic sequence if \(a_6=12\) and \(a_{14}=28\).
- Answer
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\(a_1=2\)
20) Find the first term or \(a_1\) of an arithmetic sequence if \(a_7=21\) and \(a_{15}=42\).
21) Find the first term or \(a_1\) of an arithmetic sequence if \(a_8=40\) and \(a_{123}=115\).
- Answer
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\(a_1=5\)
22) Find the first term or \(a_1\) of an arithmetic sequence if \(a_9=54\) and \(a_{17}=102\).
23) Find the first term or \(a_1\) of an arithmetic sequence if \(a_{11}=11\) and \(a_{21}=16\).
- Answer
-
\(a_1=6\)
For the exercises 24-25, find the specified term given two terms from an arithmetic sequence.
24) \(a_1=33\) and \(a_7=-15\). Find \(a_4\).
25) \(a_3=-17.1\) and \(a_{10}=-15.7\). Find \(a_{21}\).
- Answer
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\(a_{21}=-13.5\)
For the exercises 26-27, use the recursive formula to write the first five terms of the arithmetic sequence.
26) \(a_1=39; a_n=a_{n-1}-3\)
27) \(a_1=-19; a_n=a_{n-1}-1.4\)
- Answer
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\(-19,-20.4,-21.8,-23.2,-24.6\)
For the exercises 28-37, write a recursive formula for each arithmetic sequence.
28) \(a=\left \{ 40,60,80,... \right \}\)
29) \(a=\left \{ 17,26,35,... \right \}\)
- Answer
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\(a_1=17; a_n=a_{n-1}+9, n\geq 2\)
30) \(a=\left \{ -1,2,5,... \right \}\)
31) \(a=\left \{ 12,17,22,... \right \}\)
- Answer
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\(a_1=12; a_n=a_{n-1}+5, n\geq 2\)
32) \(a=\left \{ -15,-7,1,... \right \}\)
33) \(a=\left \{ 8.9,10.3,11.7,... \right \}\)
- Answer
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\(a_1=8.9; a_n=a_{n-1}+1.4, n\geq 2\)
34) \(a=\left \{ -0.52,-1.02,-1.52,... \right \}\)
35) \(a=\left \{ \dfrac{1}{5},\dfrac{9}{20},\dfrac{7}{10},... \right \}\)
- Answer
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\(a_1=\dfrac{1}{5}; a_n=a_{n-1}+\dfrac{1}{4}, n\geq 2\)
36) \(a=\left \{ -\dfrac{1}{2},-\dfrac{5}{4},-2,... \right \}\)
37) \(a=\left \{ \dfrac{1}{6},-\dfrac{11}{12},-2,... \right \}\)
- Answer
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\(a_1=\dfrac{1}{6}; a_n=a_{n-1}-\dfrac{13}{12}, n\geq 2\)
For the exercises 38-40, write a recursive formula for the given arithmetic sequence, and then find the specified term.
38) \(a=\left \{ 7, 4, 1, ... \right \}\); Find the \(17^{th}\) term.
39) \(a=\left \{ 4, 11, 18, ... \right \}\); Find the \(14^{th}\) term.
- Answer
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\(a_1=4; a_n=a_{n-1}+7, a_{14}=95\)
40) \(a=\left \{ 2, 6, 10, ... \right \}\); Find the \(12^{th}\) term.
For the exercises 41-42, use the explicit formula to write the first five terms of the arithmetic sequence.
41) \(a_n=24-4n\)
- Answer
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First five terms: \(20,16,12,8,4\).
42) \(a_n=\dfrac{1}{2}n-\dfrac{1}{2}\)
For the exercises 43-52, write an explicit formula for each arithmetic sequence.
43) \(a=\left \{ 3,5,7, ... \right \}\)
- Answer
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\(a_n=1+2n\)
44) \(a=\left \{ 32,24,16,... \right \}\)
45) \(a=\left \{ -5, 95, 195, ... \right \}\)
- Answer
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\(a_n=-105+100n\)
46) \(a=\left \{ -17, -217, -417,... \right \}\)
47) \(a=\left \{ 1.8, 3.6, 5.4, ... \right \}\)
- Answer
-
\(a_n=1.8n\)
48) \(a=\left \{ -18.1,-16.2,-14.3,... \right \}\)
49) \(a=\left \{ 15.8,18.5,21.2,... \right \}\)
- Answer
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\(a_n=13.1+2.7n\)
50) \(a=\left \{ \dfrac{1}{3},-\dfrac{4}{3},-3,... \right \}\)
51) \(a=\left \{ 0,\dfrac{1}{3},\dfrac{2}{3},... \right \}\)
- Answer
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\(a_n=\dfrac{1}{3}n-\dfrac{1}{3}\)
52) \(a=\left \{ -5,-\dfrac{10}{3},-\dfrac{5}{3},... \right \}\)
For the exercises 53-55, find the number of terms in the given finite arithmetic sequence.
53) \(a=\left \{ 3,-4,-11, ...,-60 \right \}\)
- Answer
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There are \(10\) terms in the sequence.
54) \(a=\left \{ 1.2,1.4,1.6,...,3.8 \right \}\)
55) \(a=\left \{ \dfrac{1}{2},2,\dfrac{7}{2},...,8 \right \}\)
- Answer
-
There are \(6\) terms in the sequence.
Graphical
For the exercises 56-57, determine whether the graph shown represents an arithmetic sequence.
56)
57)
- Answer
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The graph does not represent an arithmetic sequence.
For the exercises 58-60, use the information provided to graph the first 5 terms of the arithmetic sequence.
58) \(a_1=0,d=4\)
59) \(a_1=9,a_n=a_{n-1}-10\)
- Answer
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60) \(a_n=-12+5n\)
Technology
For the exercises 61-63, follow the steps to work with the arithmetic sequence \(a_n=3n-2\) using a graphing calculator:
-
Press
[MODE]
- Select SEQ in the fourth line
- Select DOT in the fifth line
- Press [ENTER]
-
Press
[Y=]
- \(n\text{Min}\) is the first counting number for the sequence. Set \(n\text{Min}=1\)
- \(u(n\) is the pattern for the sequence. Set \(u(n)=3n-2\)
- \(u(n\text{Min})\) is the first number in the sequence. Set \(u(n\text{Min})=1\)
-
Press
[2ND]
then
[WINDOW]
to go to
TBLSET
- Set \(\text{TblStart}=1\)
- Set \(\Delta \text{Tbl}=1\)
- Set Indpnt: Auto and Depend: Auto
- Press [2ND] then [GRAPH] to go to the TABLE
61) What are the first seven terms shown in the column with the heading \(u(n)\)?
- Answer
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\(1,4,7,10,13,16,19\)
62) Use the scroll-down arrow to scroll to \(n=50\)
63) Press [WINDOW] . Set \(n\text{Min}=1, n\text{Max}=5, x\text{Min}=0, x\text{Max}=6, y\text{Min}=-1, y\text{Max}=14\). Then press [GRAPH] . Graph the sequence as it appears on the graphing calculator.
- Answer
-
For the exercises 64-65, follow the steps given above to work with the arithmetic sequence \(a_n=\dfrac{1}{2}n+5\) using a graphing calculator.
64) What are the first seven terms shown in the column with the heading \(u(n)\) in the TABLE feature?
65) Graph the sequence as it appears on the graphing calculator. Be sure to adjust the WINDOW settings as needed.
- Answer
-
Extensions
66) Give two examples of arithmetic sequences whose \(4^{th}\) terms are \(9\).
67) Give two examples of arithmetic sequences whose \(10^{th}\) terms are \(206\).
- Answer
-
Answers will vary. Examples: \(a_n=20.6n\) and \(a_n=2+20.4n\).
68) Find the \(5^{th}\) term of the arithmetic sequence \(\left \{ 9b,5b,b, ... \right \}\).
69) Find the \( 11^{ th}\) term of the arithmetic sequence \(\left \{ 3a-2b,a+2b,-a+6b ... \right \}\)
- Answer
-
\(a_{11}=-17a+38b\)
70) At which term does the sequence \(\left \{ 5.4,14.5,23.6,... \right \}\) exceed \(151\)?
71) At which term does the sequence \(\left \{ \dfrac{17}{3},\dfrac{31}{6},\dfrac{14}{3},... \right \}\) begin to have negative values?
- Answer
-
The sequence begins to have negative values at the \(13^{th}\) term, \(a_{13}=-\dfrac{1}{3}\)
72) For which terms does the finite arithmetic sequence \(\left \{ \dfrac{5}{2},\dfrac{19}{8},\dfrac{9}{4},...,\dfrac{1}{8} \right \}\) have integer values?
73) Write an arithmetic sequence using a recursive formula. Show the first \(4\) terms, and then find the \(31^{st}\) term.
- Answer
-
Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: \(a_1=3,a_n=a_{n-1}-3\). First \(4\) terms: \(3,0,-3,-6, a_{31}=-87\)
74) Write an arithmetic sequence using an explicit formula. Show the first \(4\) terms, and then find the \(28^{th}\) term.
11.3 Geometric Sequences
Verbal
1) What is a geometric sequence?
- Answer
-
A sequence in which the ratio between any two consecutive terms is constant.
2) How is the common ratio of a geometric sequence found?
3) What is the procedure for determining whether a sequence is geometric?
- Answer
-
Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.
4) What is the difference between an arithmetic sequence and a geometric sequence?
5) Describe how exponential functions and geometric sequences are similar. How are they different?
- Answer
-
Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.
Algebraic
For the exercises 6-8, find the common ratio for the geometric sequence.
6) \(1,3,9,27,81,...\)
7) \(-0.125,0.25,-0.5,1,-2,...\)
- Answer
-
The common ratio is \(-2\)
8) \(-2,-\dfrac{1}{2},-\dfrac{1}{8},-\dfrac{1}{32},-\dfrac{1}{128},...\)
For the exercises 9-13, determine whether the sequence is geometric. If so, find the common ratio.
9) \(-6,-12,-24,-48,-96,...\)
- Answer
-
The sequence is geometric. The common ratio is \(2\).
10) \(5,5.2,5.4,5.6,5.8,...\)
11) \(-1,\dfrac{1}{2},-\dfrac{1}{4},\dfrac{1}{8},-\dfrac{1}{16},...\)
- Answer
-
The sequence is geometric. The common ratio is \(-\dfrac{1}{2}\).
12) \(6,8,11,15,20,...\)
13) \(0.8,4,20,100,500,...\)
- Answer
-
The sequence is geometric. The common ratio is \(5\).
For the exercises 14-15, write the first five terms of the geometric sequence, given the first term and common ratio.
14) \(a_1=8,r=0.3\)
15) \(a_1=5,r=\dfrac{1}{5}\)
- Answer
-
\(5,1,\dfrac{1}{5},\dfrac{1}{25},\dfrac{1}{125}\)
For the exercises 16-17, write the first five terms of the geometric sequence, given any two terms.
16) \(a_7=64, a_{10}=512\)
17) \(a_6=25, a_8=6.25\)
- Answer
-
\(800,400,200,100,50\)
For the exercises 18-19, find the specified term for the geometric sequence, given the first term and common ratio.
18) The first term is \(2\), and the common ratio is \(3\). Find the \(5^{th}\) term.
19) The first term is \(16\) and the common ratio is \(-\dfrac{1}{3}\). Find the \(4^{th}\) term.
- Answer
-
\(a_4=-\dfrac{16}{27}\)
For the exercises 20-21, find the specified term for the geometric sequence, given the first four terms.
20) \(a_n=\left \{ -1,2,-4,8,... \right \}\). Find \(a_{12}\).
21) \(a_n=\left \{ -2,\dfrac{2}{3},-\dfrac{2}{9},\dfrac{2}{27},... \right \}\). Find \(a_7\).
- Answer
-
\(a_7=-\dfrac{2}{729}\)
For the exercises 22-23, write the first five terms of the geometric sequence.
22) \(a_1=-486, a_n=-\dfrac{1}{3}a_{n-1}\)
23) \(a_1=7, a_n=0.2a_{n-1}\)
- Answer
-
\(7,1.4,0.28,0.056,0.0112\)
For the exercises 24-31, write a recursive formula for each geometric sequence.
24) \(a_n=\left \{ -1,5,-25,125,... \right \}\)
25) \(a_n=\left \{ -32,-16,-8,-4,... \right \}\)
- Answer
-
\(a_1=-32, a_n=\dfrac{1}{2}a_{n-1}\)
26) \(a_n=\left \{ 14,56,224,896,... \right \}\)
27) \(a_n=\left \{ 10,-3,0.9,-0.27,... \right \}\)
- Answer
-
\(a_1=10, a_n=-0.3a_{n-1}\)
28) \(a_n=\left \{ 0.61,1.83,5.49,16.47,... \right \}\)
29) \(a_n=\left \{ \dfrac{3}{5},\dfrac{1}{10},\dfrac{1}{60},\dfrac{1}{360},... \right \}\)
- Answer
-
\(a_1=\dfrac{3}{5}, a_n=\dfrac{1}{26}a_{n-1}\)
30) \(a_n=\left \{ -2,\dfrac{4}{3},-\dfrac{8}{9},\dfrac{16}{27},... \right \}\)
31) \(a_n=\left \{ \dfrac{1}{512},-\dfrac{1}{128},\dfrac{1}{32},-\dfrac{1}{8},... \right \}\)
- Answer
-
\(a_1=\dfrac{1}{512}, a_n=-4a_{n-1}\)
For the exercises 32-33, write the first five terms of the geometric sequence.
32) \(a_n=-4\cdot 5^{n-1}\)
33) \(a_n=12\cdot \left ( -\dfrac{1}{2} \right )^{n-1}\)
- Answer
-
\(12,-2,3,-\dfrac{3}{2} ,\dfrac{3}{4}\)
For the exercises 34-41, write an explicit formula for each geometric sequence.
34) \(a_n=\left \{ -2,-4,-8,-16,... \right \}\)
35) \(a_n=\left \{ 1,3,9,27,... \right \}\)
- Answer
-
\(a_n=3^{n-1}\)
36) \(a_n=\left \{ -4,-12,-36,-108,... \right \}\)
37) \(a_n=\left \{ 0.8,-4,20,-100,... \right \}\)
- Answer
-
\(a_n=0.8\cdot (-5)^{n-1}\)
38) \(a_n=\left \{ -1.25,-5,-20,-80,... \right \}\)
39) \(a_n=\left \{ -1,-\dfrac{4}{5},-\dfrac{16}{25},-\dfrac{64}{125},... \right \}\)
- Answer
-
\(a_n=-\left ( \dfrac{4}{5} \right )^{n-1}\)
40) \(a_n=\left \{ 2,\dfrac{1}{3},\dfrac{1}{18},\dfrac{1}{108},... \right \}\)
41) \(a_n=\left \{ 3,-1,\dfrac{1}{3},-\dfrac{1}{9},... \right \}\)
- Answer
-
\(a_n=3\cdot \left ( -\dfrac{1}{3} \right )^{n-1}\)
For the exercises 42-43, find the specified term for the geometric sequence given.
42) Let \(a_1=4\), \(a_n=-3a_{n-1}\). Find \(a_8\).
43) Let \(a_n=-\left ( -\dfrac{1}{3} \right )^{n-1}\). Find \(a_{12}\).
- Answer
-
\(a_{12}=\dfrac{1}{177,147}\)
For the exercises 44-45, find the number of terms in the given finite geometric sequence.
44) \(a_n=\left \{ -1,3,-9,...,2187 \right \}\)
45) \(a_n=\left \{ 2,1,\dfrac{1}{2},...,\dfrac{1}{1024} \right \}\)
- Answer
-
There are \(12\) terms in the sequence.
Graphical
For the exercises 46-47, determine whether the graph shown represents a geometric sequence.
46)
47)
- Answer
-
The graph does not represent a geometric sequence.
For the exercises 48-50, use the information provided to graph the first five terms of the geometric sequence.
48) \(a_1=1, r=\dfrac{1}{2}\)
49) \(a_1=3, a_n=2a_{n-1}\)
- Answer
-
50) \(a_n=27\cdot 0.3^{n-1}\)
Extensions
51) Use recursive formulas to give two examples of geometric sequences whose \(3^{rd}\) terms are \(200\).
- Answer
-
Answers will vary. Examples: \(a_1=800, a_n=0.5a_{n-1}\) and \(a_1=12.5, a_n=4a_{n-1}\)
52) Use explicit formulas to give two examples of geometric sequences whose \(7^{th}\) terms are \(1024\).
53) Find the \(5^{th}\) term of the geometric sequence \(\left \{ b,4b,16b,... \right \}\).
- Answer
-
\(a_5=256b\)
54) Find the \(7^{th}\) term of the geometric sequence \(\left \{ 64a(-b),32a(-3b),16a(-9b),... \right \}\).
55) At which term does the sequence \(\left \{ 10,12,14.4,17.28, ... \right \}\) exceed \(100\)?
- Answer
-
The sequence exceeds \(100\) at the \(14^{th}\) term, \(a_{14} \approx 107\).
56) At which term does the sequence \(\left \{ \dfrac{1}{2187},\dfrac{1}{729},\dfrac{1}{243},\dfrac{1}{81}, ... \right \}\) begin to have integer values?
57) For which term does the geometric sequence \(a_n=-36\left (\dfrac{2}{3} \right )^{n-1}\) first have a non-integer value?
- Answer
-
\(a_4=-\dfrac{32}{3}\)
58) Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the \(10^{th}\) term.
59) Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between \(0\) and \(1\). Show the first \(4\) terms, and then find the \(8^{th}\) term.
- Answer
-
Answers will vary. Example: Explicit formula with a decimal common ratio: \(a_n=400\cdot 0.5^{n-1}\); First \(4\) terms: \(400,200,100,50; a_8=3.125\)
60) Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.
11.4 Series and Their Notations
Verbal
1) What is an \(n\text{th} \) partial sum?
- Answer
-
An \(n\text{th} \) partial sum is the sum of the first \(n\) terms of a sequence.
2) What is the difference between an arithmetic sequence and an arithmetic series?
3) What is a geometric series?
- Answer
-
A geometric series is the sum of the terms in a geometric sequence.
4) How is finding the sum of an infinite geometric series different from finding the \(n\text{th} \) partial sum?
5) What is an annuity?
- Answer
-
An annuity is a series of regular equal payments that earn a constant compounded interest.
Algebraic
For the exercises 6-9, express each description of a sum using summation notation.
6) The sum of terms \(m^2+3m\) from \(m=1\) to \(m=5\).
7) The sum from of \(n=0\) to \(n=4\) of \(5n\)
- Answer
-
\(\displaystyle \sum_{n=0}^{4}5n\)
8) The sum of \(6k-5\) from \(k=-2\) to \(k-1\)
9) The sum that results from adding the number \(4\) five times
- Answer
-
\(\displaystyle \sum_{k=1}^{5}4\)
For the exercises 10-12, express each arithmetic sum using summation notation.
10) \(5+10+15+20+25+30+35+40+45+50\)
11) \(10+18+26+\ldots +162\)
- Answer
-
\(\displaystyle \sum_{k=1}^{20} 8k+2\)
12) \(\dfrac{1}{2}+1+\dfrac{3}{2}+2+\ldots +4\)
For the exercises 13-15, use the formula for the sum of the first \(n\) terms of each arithmetic sequence.
13) \(\dfrac{3}{2}+2+\dfrac{5}{2}+3+\dfrac{7}{2}\)
- Answer
-
\(S_5=\dfrac{5\left ( \tfrac{3}{2}+\tfrac{7}{2} \right )}{2}\)
14) \(19+25+31+\ldots +73\)
15) \(3.2+3.4+3.6+\ldots +5.6\)
- Answer
-
\(S_{13}=\dfrac{13\left ( 3.2+5.6 \right )}{2}\)
For the exercises 16-18, express each geometric sum using summation notation.
16) \(1+3+9+27+81+243+729+2187\)
17) \(8+4+2+\ldots +0.125\)
- Answer
-
\(\displaystyle \sum_{k=1}^{7}8\cdot 0.5^{k-1}\)
18) \(-\dfrac{1}{6}+\dfrac{1}{12}-\dfrac{1}{24}+\ldots +\dfrac{1}{768}\)
For the exercises 19-21, use the formula for the sum of the first \(n\) terms of each geometric sequence, and then state the indicated sum.
19) \(9+3+1+\dfrac{1}{3}+\dfrac{1}{9}\)
- Answer
-
\(S_5=\dfrac{9\left ( 1-\left (\tfrac{1}{3} \right )^5 \right )}{1-\tfrac{1}{3}}=\dfrac{121}{9}\approx 13.44\)
20) \(\displaystyle \sum_{n=1}^{9}5\cdot 2^{n-1}\)
21) \(\displaystyle \sum_{a=1}^{11}64\cdot 0.2^{a-1}\)
- Answer
-
\(S_{11}=\dfrac{64(1-0.2^{11})}{1-0.2}=\dfrac{781,249,984}{9,765,625}\approx 80\)
For the exercises 22-25, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.
22) \(12+18+24+30+\ldots\)
23) \(2+1.6+1.28+1.024+\ldots\)
- Answer
-
The series is defined. \(S=\dfrac{2}{1-0.8}\)
24) \(\displaystyle \sum_{m=1}^{\infty }4^{m-1}\)
25) \(\sum_{\infty }^{k=1} -\left ( -\frac{1}{2} \right )^{k-1}\)
- Answer
-
The series is defined. \(S=\dfrac{-1}{1-\left ( -\tfrac{1}{2} \right )}\)
Graphical
For the exercises 26-27, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of \(\$50\). Each month thereafter he increased the previous deposit amount by \(\$20\).
26) Graph the arithmetic sequence showing one year of Javier’s deposits.
27) Graph the arithmetic series showing the monthly sums of one year of Javier’s deposits.
- Answer
-
For the exercises 28-29, use the geometric series \(\displaystyle \sum_{k=1}^{\infty }\left ( \dfrac{1}{2} \right )^k\)
28) Graph the first \(7\) partial sums of the series.
29) What number does \(S_n\) seem to be approaching in the graph? Find the sum to explain why this makes sense.
- Answer
-
Sample answer: The graph of \(S_n\) seems to be approaching \(1\). This makes sense because \(\displaystyle \sum_{k=1}^{\infty }\left ( \dfrac{1}{2} \right )^k\) is a defined infinite geometric series with \(S=\dfrac{\tfrac{1}{2}}{1-\left ( \tfrac{1}{2} \right )}=1\).
Numeric
For the exercises 30-33, find the indicated sum.
30) \(\displaystyle \sum_{a=1}^{14}a\)
31) \(\displaystyle \sum_{n=1}^{6}n(n-2)\)
- Answer
-
\(49\)
32) \(\displaystyle \sum_{k=1}^{17}k^2\)
33) \(\displaystyle \sum_{k=1}^{7}2^k\)
- Answer
-
\(254\)
For the exercises 34-37, use the formula for the sum of the first \(n\) terms of an arithmetic series to find the sum.
34) \(-1.7+-0.4+0.9+2.2+3.5+4.8\)
35) \(6+\dfrac{15}{2}+9+\dfrac{21}{2}+12+\dfrac{27}{2}+15\)
- Answer
-
\(S_7=\dfrac{147}{2}\)
36) \(-1+3+7+\ldots +31\)
37) \(\displaystyle \sum_{k=1}^{11}\left ( \dfrac{k}{2}-\dfrac{1}{2} \right )\)
- Answer
-
\(S_{11}=\dfrac{55}{2}\)
For the exercises 38-41, use the formula for the sum of the first \(n\) terms of a geometric series to find the partial sum.
38) \(S_6\) for the series \(-2-10-50-250\ldots\)
39) \(S_7\) for the series \(0.4-2+10-50\ldots\)
- Answer
-
\(S_7=5208.4\)
40) \(\displaystyle \sum_{k=1}^{9}2^{k-1}\)
41) \(\displaystyle \sum_{n=1}^{10}-2\cdot \left ( \dfrac{1}{2} \right )^{n-1}\)
- Answer
-
\(S_{10}=-\dfrac{1023}{256}\)
For the exercises 42-45, find the sum of the infinite geometric series.
42) \(4+2+1+\dfrac{1}{2}+\ldots\)
43) \(-1-\dfrac{1}{4}-\dfrac{1}{16}-\dfrac{1}{64}\ldots\)
- Answer
-
\(S=-\dfrac{4}{3}\)
44) \(\sum_{\infty }^{k=1}3\cdot \left ( \dfrac{1}{4} \right )^{k-1}\)
45) \(\displaystyle \sum_{n=1}^{\infty }4.6\cdot 0.5^{n-1}\)
- Answer
-
\(S=9.2\)
For the exercises 46-49, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.
46) Deposit amount: \(\$50\); total deposits: \(60\); interest rate: \(5\%\), compounded monthly
47) Deposit amount: \(\$150\); total deposits: \(24\); interest rate: \(3\%\), compounded monthly
- Answer
-
\(\$3705.42\)
48) Deposit amount: \(\$450\); total deposits: \(60\); interest rate: \(4.5\%\), compounded quarterly
49) Deposit amount: \(\$100\); total deposits: \(120\); interest rate: \(10\%\), compounded semi-annually
- Answer
-
\(\$695,823.97\)
Extensions
50) The sum of terms \(50-k^2\) from \(k=x\) through \(7\) is \(115\). What is \(x\)?
51) Write an explicit formula for \(a_k\) such that \(\displaystyle \sum_{k=0}^{6}a_k=189\). Assume this is an arithmetic series.
- Answer
-
\(a_k=30-k\)
52) Find the smallest value of \(n\) such that \(\displaystyle \sum_{k=1}^{n}(3k-5)>100\).
53) How many terms must be added before the series \(-1-3-5-7\ldots\) has a sum less than \(-75\)?
- Answer
-
\(9\) terms
54) Write \(0.\overline{65}\) as an infinite geometric series using summation notation. Then use the formula for finding the sum of an infinite geometric series to convert \(0.\overline{65}\) to a fraction.
55) The sum of an infinite geometric series is five times the value of the first term. What is the common ratio of the series?
- Answer
-
\(r=\dfrac{4}{5}\)
56) To get the best loan rates available, the Riches want to save enough money to place \(20\%\) down on a \(\$160,000\) home. They plan to make monthly deposits of \(\$125\) in an investment account that offers \(8.5\%\) annual interest compounded semi-annually. Will the Riches have enough for a \(20\%\) down payment after five years of saving? How much money will they have saved?
57) Karl has two years to save \(\$10,000\) to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a \(4.2\%\) annual interest rate that compounds monthly?
- Answer
-
\(\$400\) per month
Real-World Applications
58) Keisha devised a week-long study plan to prepare for finals. On the first day, she plans to study for \(1\) hour, and each successive day she will increase her study time by \(30\) minutes. How many hours will Keisha have studied after one week?
59) A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by \(8\) feet. How far did the boulder travel after \(10\) seconds?
- Answer
-
\(420\) feet
60) A scientist places \(50\) cells in a petri dish. Every hour, the population increases by \(1.5\%\). What will the cell count be after \(1\) day?
61) A pendulum travels a distance of \(3\) feet on its first swing. On each successive swing, it travels \(\dfrac{3}{4}\) the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?
- Answer
-
\(12\) feet
62) Rachael deposits \(\$1,500\) into a retirement fund each year. The fund earns \(8.2\%\) annual interest, compounded monthly. If she opened her account when she was \(19\) years old, how much will she have by the time she is \(55\)? How much of that amount will be interest earned?
11.5 Counting Principles
Verbal
For the exercises 1-2, assume that there are \(n\) ways an event \(A\) can happen, \(m\) ways an event \(B\) can happen, and that \(A\) and \(B\) are non-overlapping.
1) Use the Addition Principle of counting to explain how many ways event \(A\) or \(B\) can occur.
- Answer
-
There are \(m+n\) ways for either event \(A\) or event \(B\) to occur.
2) Use the Multiplication Principle of counting to explain how many ways event \(A\) and \(B\) can occur.
Answer the questions 3-5.
3) When given two separate events, how do we know whether to apply the Addition Principle or the Multiplication Principle when calculating possible outcomes? What conjunctions may help to determine which operations to use?
- Answer
-
The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem.
4) Describe how the permutation of \(n\) objects differs from the permutation of choosing \(r\) objects from a set of \(n\) objects. Include how each is calculated.
5) What is the term for the arrangement that selects \(r\) objects from a set of \(n\) objects when the order of the \(r\) objects is not important? What is the formula for calculating the number of possible outcomes for this type of arrangement?
- Answer
-
A combination; \(C(n,r)=\dfrac{n!}{(n-r)!r!}\)
Numeric
For the exercises 6-14, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.
6) Let the set \(A=\left \{ -5,-3,-1,2,3,4,5,6 \right \}\). How many ways are there to choose a negative or an even number from \(A\)?
7) Let the set \(B=\left \{ -23,-16,-7,-2,20,36,48,72 \right \}\). How many ways are there to choose a positive or an odd number from \(A\)?
- Answer
-
\(4+2=6\)
8) How many ways are there to pick a red ace or a club from a standard card playing deck?
9) How many ways are there to pick a paint color from \(5\) shades of green, \(4\) shades of blue, or \(7\) shades of yellow?
- Answer
-
\(5+4+7=16\)
10) How many outcomes are possible from tossing a pair of coins?
11) How many outcomes are possible from tossing a coin and rolling a \(6\)-sided die?
- Answer
-
\(2\times 6=12\)
12) How many two-letter strings—the first letter from \(A\) and the second letter from \(B\) — can be formed from the sets \(A=\left \{ b,c,d \right \}\) and \(B=\left \{ a,e,i,o,u \right \}\)?
13) How many ways are there to construct a string of \(3\) digits if numbers can be repeated?
- Answer
-
\(10^3=1000\)
14) How many ways are there to construct a string of \(3\) digits if numbers cannot be repeated?
For the exercises 15-24, compute the value of the expression.
15) \(P(5,2)\)
- Answer
-
\(P(5,2)=20\)
16) \(P(8,4)\)
17) \(P(3,3)\)
- Answer
-
\(P(3,3)=6\)
18) \(P(9,6)\)
19) \(P(11,5)\)
- Answer
-
\(P(11,5)=55,440\)
20) \(C(8,5)\)
21) \(C(12,4)\)
- Answer
-
\(C(12,4)=495\)
22) \(C(26,3)\)
23) \(C(7,6)\)
- Answer
-
\(C(7,6)=7\)
24) \(C(10,3)\)
For the exercises 25-29, find the number of subsets in each given set.
25) \(\left \{ 1,2,3,4,5,6,7,8,9,10 \right \}\)
- Answer
-
\(2^{10}=1024\)
26) \(\left \{ a,b,c,\ldots ,z \right \}\)
27) A set containing \(5\) distinct numbers, \(4\) distinct letters, and \(3\) distinct symbols
- Answer
-
\(2^{12}=4096\)
28) The set of even numbers from \(2\) to \(28\)
29) The set of two-digit numbers between \(1\) and \(100\) containing the digit \(0\)
- Answer
-
\(2^{9}=512\)
For the exercises 30-34, find the distinct number of arrangements.
30) The letters in the word “juggernaut”
31) The letters in the word “academia”
- Answer
-
\(\dfrac{8!}{3!}=6720\)
32) The letters in the word “academia” that begin and end in “a”
33) The symbols in the string \(\#,\#,\#,@,@,\$,\$,\$,\%,\%,\%,\%\)
- Answer
-
\(\dfrac{12!}{3!2!3!4!}\)
34) The symbols in the string \(\#,\#,\#,@,@,\$,\$,\$,\%,\%,\%,\%\) that begin and end with “\(\%\)”
Extensions
35) The set, \(S\) consists of \(900,000,000\) whole numbers, each being the same number of digits long. How many digits long is a number from \(S\)? ( Hint: use the fact that a whole number cannot start with the digit \(0\).)
- Answer
-
\(9\)
36) The number of \(5\)-element subsets from a set containing n
37) Can \(C(n,r)\) ever equal \(P(n,r)\)? Explain.
- Answer
-
Yes, for the trivial cases \(r=0\) and \(r=1\). If \(r=0\), then \(C(n,r)=P(n,r)=1\). If \(r=1\), then \(r=1\), \(C(n,r)=P(n,r)=n\).
38) Suppose a set \(A\) has \(2,048\) subsets. How many distinct objects are contained in \(A\)?
39) How many arrangements can be made from the letters of the word “mountains” if all the vowels must form a string?
- Answer
-
\(\dfrac{6!}{2!}\times 4!=8640\)
Real-World Applications
40) A family consisting of \(2\) parents and \(3\) children is to pose for a picture with \(2\) family members in the front and \(3\) in the back.
- How many arrangements are possible with no restrictions?
- How many arrangements are possible if the parents must sit in the front?
- How many arrangements are possible if the parents must be next to each other?
41) A cell phone company offers \(6\) different voice packages and \(8\) different data packages. Of those, \(3\) packages include both voice and data. How many ways are there to choose either voice or data, but not both?
- Answer
-
\(6-3+8-3=8\)
42) In horse racing, a “trifecta” occurs when a bettor wins by selecting the first three finishers in the exact order (\(1^{st}\) place, \(2^{nd}\) place, and \(3^{rd}\) place). How many different trifectas are possible if there are \(14\) horses in a race?
43) A wholesale T-shirt company offers sizes small, medium, large, and extra-large in organic or non-organic cotton and colors white, black, gray, blue, and red. How many different T-shirts are there to choose from?
- Answer
-
\(4\times 2\times 5=40\)
44) Hector wants to place billboard advertisements throughout the county for his new business. How many ways can Hector choose \(15\) neighborhoods to advertise in if there are \(30\) neighborhoods in the county?
45) An art store has \(4\) brands of paint pens in \(12\) different colors and \(3\) types of ink. How many paint pens are there to choose from?
- Answer
-
\(4\times 12\times 3=144\)
46) How many ways can a committee of \(3\) freshmen and \(4\) juniors be formed from a group of \(8\) freshmen and \(11\) juniors?
47) How many ways can a baseball coach arrange the order of \(9\) batters if there are \(15\) players on the team?
- Answer
-
\(P(15,9)=1,816,214,400\)
48) A conductor needs \(5\) cellists and \(5\) violinists to play at a diplomatic event. To do this, he ranks the orchestra’s \(10\) cellists and \(16\) violinists in order of musical proficiency. What is the ratio of the total cellist rankings possible to the total violinist rankings possible?
49) A motorcycle shop has \(10\) choppers, \(6\) bobbers, and \(5\) café racers—different types of vintage motorcycles. How many ways can the shop choose \(3\) choppers, \(5\) bobbers, and \(2\) café racers for a weekend showcase?
- Answer
-
\(C(10,3)\times C(6,5)\times C(5,2)=7,200\)
50) A skateboard shop stocks \(10\) types of board decks, \(3\) types of trucks, and \(4\) types of wheels. How many different skateboards can be constructed?
51) Just-For-Kicks Sneaker Company offers an online customizing service. How many ways are there to design a custom pair of Just-For-Kicks sneakers if a customer can choose from a basic shoe up to \(11\) customizable options?
- Answer
-
\(2^{11}=2048\)
52) A car wash offers the following optional services to the basic wash: clear coat wax, triple foam polish, undercarriage wash, rust inhibitor, wheel brightener, air freshener, and interior shampoo. How many washes are possible if any number of options can be added to the basic wash?
53) Susan bought \(20\) plants to arrange along the border of her garden. How many distinct arrangements can she make if the plants are comprised of \(6\) tulips, \(6\) roses, and \(8\) daisies?
- Answer
-
\(\dfrac{20!}{6!6!8!}=116,396,280\)
54) How many unique ways can a string of Christmas lights be arranged from \(9\) red, \(10\) green, \(6\) white, and \(12\) gold color bulbs?
11.6 Binomial Theorem
Verbal
1) What is a binomial coefficient, and how it is calculated?
- Answer
-
A binomial coefficient is an alternative way of denoting the combination \(C(n,r)\). It is defined as \(\dbinom{n}{r}=C(n,r)=\dfrac{n!}{r!(n-r)!}\).
2) What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?
3) What is the Binomial Theorem and what is its use?
- Answer
-
The Binomial Theorem is defined as \((x+y)^n=\displaystyle \sum_{k=0}^{n}\dbinom{n}{r}x^{n-k}y^k\) and can be used to expand any binomial.
4) When is it an advantage to use the Binomial Theorem? Explain.
Algebraic
For the exercises 5-12, evaluate the binomial coefficient.
5) \(\dbinom{6}{2}\)
- Answer
-
\(15\)
6) \(\dbinom{5}{3}\)
7) \(\dbinom{7}{4}\)
- Answer
-
\(35\)
8) \(\dbinom{9}{7}\)
9) \(\dbinom{10}{9}\)
- Answer
-
\(10\)
10) \(\dbinom{25}{11}\)
11) \(\dbinom{17}{6}\)
- Answer
-
\(12,376\)
12) \(\dbinom{200}{199}\)
For the exercises 13-22, use the Binomial Theorem to expand each binomial.
13) \((4a-b)^3\)
- Answer
-
\(64a^3-48a^2b+12ab^2-b^3\)
14) \((5a+2)^3\)
15) \((3a+2b)^3\)
- Answer
-
\(27a^3+54a^2b+36ab^2+8b^3\)
16) \((2x+3y)^4\)
17) \((4x+2y)^5\)
- Answer
-
\(1024x^5+2560x^4y+2560x^3y^2+1280x^2y^3+320xy^4+32y^5\)
18) \((3x-2y)^4\)
19) \((4x-3y)^5\)
- Answer
-
\(1024x^5-3840x^4y+5760x^3y^2-4320x^2y^3+1620xy^4-243y^5\)
20) \(\left ( \dfrac{1}{x}+3y \right )^5\)
21) \(\left ( x^{-1}+2y^{-1} \right )^4\)
- Answer
-
\(\dfrac{1}{x^4}+\dfrac{8}{x^3y}+\dfrac{24}{x^2y^2}+\dfrac{32}{xy^3}+\dfrac{16}{y^4}\)
22) \(\left ( \sqrt{x}-\sqrt{y} \right )^5\)
For the exercises 23-29, use the Binomial Theorem to write the first three terms of each binomial.
23) \((a+b)^{17}\)
- Answer
-
\(a^{17}+17a^{16}b+136a^{15}b^2\)
24) \((x-1)^{18}\)
25) \((a-2b)^{15}\)
- Answer
-
\(a^{15}-30a^{14}b+420a^{13}b^2\)
26) \((x-2y)^8\)
27) \((3a+b)^{20}\)
- Answer
-
\(3,486,784,401a^{20}+23,245,229,340a^{19}b+73,609,892,910a^{18}b^2\)
28) \((2a+4b)^7\)
29) \(\left ( x^3-\sqrt{y} \right )^8\)
- Answer
-
\(x^{24}-8x^{21}\sqrt{y}+28x^{18}y\)
For the exercises 30-39, find the indicated term of each binomial without fully expanding the binomial.
30) The fourth term of \((2x-3y)^4\)
31) The fourth term of \((3x-2y)^5\)
- Answer
-
\(-720x^2y^3\)
32) The third term of \((6x-3y)^7\)
33) The eighth term of \((7+5y)^{14}\)
- Answer
-
\(220,812,466,875,000y^7\)
34) The seventh term of \((a+b)^{11}\)
35) The fifth term of \((x-y)^{7}\)
- Answer
-
\(35x^3y^4\)
36) The tenth term of \((x-1)^{12}\)
37) The ninth term of \((a-3b^2)^{11}\)
- Answer
-
\(1,082,565a^3b^{16}\)
38) The fourth term of \(\left ( x^3-\dfrac{1}{2} \right )^{10}\)
39) The eighth term of \(\left ( \dfrac{y}{2}+\dfrac{2}{x} \right )^{9}\)
- Answer
-
\(\dfrac{1152y^2}{x^7}\)
Graphical
For the exercises 40-44, use the Binomial Theorem to expand the binomial \(f(x)=(x+3)^4\). Then find and graph each indicated sum on one set of axes.
40) Find and graph \(f_1(x)\), such that \(f_1(x)\) is the first term of the expansion.
41) Find and graph \(f_2(x)\), such that \(f_2(x)\) is the sum of the first two terms of the expansion.
- Answer
-
\(f_2(x)=x^4+12x^3\)
42) Find and graph \(f_3(x)\), such that \(f_3(x)\) is the sum of the first three terms of the expansion.
43) Find and graph \(f_4(x)\), such that \(f_4(x)\) is the sum of the first four terms of the expansion.
- Answer
-
\(f_4(x)=x^4+12x^3+54x^2+108x\)
44) Find and graph \(f_5(x)\), such that \(f_5(x)\) is the sum of the first five terms of the expansion.
Extensions
45) In the expansion of \((5x+3y)^n\), each term has the form \(\dbinom{n}{k}a^{n-k}b^k\) where \(k\) successively takes on the value \(0,1,2,\ldots ,n\). If \(\dbinom{n}{k}=\dbinom{7}{2}\), what is the corresponding term?
- Answer
-
\(590,625x^5y^2\)
46) In the expansion of \((a+b)^n\), the coefficient of \(a^{n-k}b^k\) is the same as the coefficient of which other term?
47) Consider the expansion of \((x+b)^{40}\). What is the exponent of \(b\) in the \(k\text {th}\) term?
- Answer
-
\(k-1\)
48) Find \(\dbinom{n}{k-1}+\dbinom{n}{k}\) and write the answer as a binomial coefficient in the form \(\dbinom{n}{k}\). Prove it. Hint: Use the fact that, for any integer \(p\), such that \(p\geq 1, p!=p(p-1)!\).
- Answer
-
\(\begin{align*}
\dbinom{n}{k-1}+\dbinom{n}{k} &= \dfrac{n!}{k!(n-k)!}+\frac{n!}{(k-1)!(n-(k-1))!}\\
&= \dfrac{n!}{k!(n-k)!}+\frac{n!}{(k-1)!(n-k+1)!}\\
&= \dfrac{(n-k+1)n!}{(n-k+1)k!(n-k)!}+\frac{kn!}{k(k-1)!(n-k+1)!}\\
&= \dfrac{(n-k+1)n!+kn!}{k!(n-k+1)!}\\
&= \dfrac{(n+1)n!}{k!((n+1)-k)!}\\
&= \dfrac{(n+1)!}{k!((n+1)-k)!}\\
&= \dbinom{n+1}{k}
\end{align*}\)
49) Which expression cannot be expanded using the Binomial Theorem? Explain.
- \((x^2-2x+1)\)
- \(\left ( \sqrt{a}+4\sqrt{a}-5 \right )^8\)
- \((x^3+2y^2-z)^5\)
- \(\left ( 3x^2-\sqrt{2y^3} \right )^{12}\)
- Answer
-
The expression \((x^3+2y^2-z)^5\) cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.
11.7 Probability
Verbal
1) What term is used to express the likelihood of an event occurring? Are there restrictions on its values? If so, what are they? If not, explain.
- Answer
-
probability; The probability of an event is restricted to values between \(0\) and \(1\), inclusive of \(0\) and \(1\).
2) What is a sample space?
3) What is an experiment?
- Answer
-
An experiment is an activity with an observable result.
4) What is the difference between events and outcomes? Give an example of both using the sample space of tossing a coin \(50\) times.
5) The union of two sets is defined as a set of elements that are present in at least one of the sets. How is this similar to the definition used for the union of two events from a probability model? How is it different?
- Answer
-
The probability of the union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets \(A\) and \(B\) and a union of events \(A\) and \(B\), the union includes either \(A\) or \(B\) or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is always a numerical value between \(0\) and \(1\).
Numeric
For the exercises 6-13, use the spinner shown in the Figure below to find the probabilities indicated.
6) Landing on red
7) Landing on a vowel
- Answer
-
\(\dfrac{1}{2}\)
8) Not landing on blue
9) Landing on purple or a vowel
- Answer
-
\(\dfrac{5}{8}\)
10) Landing on blue or a vowel
11) Landing on green or blue
- Answer
-
\(\dfrac{1}{2}\)
12) Landing on yellow or a consonant
13) Not landing on yellow or a consonant
- Answer
-
\(\dfrac{3}{8}\)
For the exercises 14-17, two coins are tossed.
14) What is the sample space?
15) Find the probability of tossing two heads.
- Answer
-
\(\dfrac{1}{4}\)
16) Find the probability of tossing exactly one tail.
17) Find the probability of tossing at least one tail.
- Answer
-
\(\dfrac{3}{4}\)
For the exercises 18-25, four coins are tossed.
18) What is the sample space?
19) Find the probability of tossing exactly two heads.
- Answer
-
\(\dfrac{3}{8}\)
20) Find the probability of tossing exactly three heads.
21) Find the probability of tossing four heads or four tails.
- Answer
-
\(\dfrac{1}{8}\)
22) Find the probability of tossing all tails.
23) Find the probability of tossing not all tails.
- Answer
-
\(\dfrac{15}{16}\)
24) Find the probability of tossing exactly two heads or at least two tails.
25) Find the probability of tossing either two heads or three heads.
- Answer
-
\(\dfrac{5}{8}\)
For the exercises 26-32, one card is drawn from a standard deck of \(52\) cards. Find the probability of drawing the following:
26) A club
27) A two
- Answer
-
\(\dfrac{1}{13}\)
28) Six or seven
29) Red six
- Answer
-
\(\dfrac{1}{26}\)
30) An ace or a diamond
31) A non-ace
- Answer
-
\(\dfrac{12}{13}\)
32) A heart or a non-jack
For the exercises 33-42, two dice are rolled, and the results are summed.
33) Construct a table showing the sample space of outcomes and sums.
- Answer
-
1 2 3 4 5 6 1 (1, 1)
2(1, 2)
3(1, 3)
4(1, 4)
5(1, 5)
6(1, 6)
72 (2, 1)
3(2, 2)
4(2, 3)
5(2, 4)
6(2, 5)
7(2, 6)
83 (3, 1)
4(3, 2)
5(3, 3)
6(3, 4)
7(3, 5)
8(3, 6)
94 (4, 1)
5(4, 2)
6(4, 3)
7(4, 4)
8(4, 5)
9(4, 6)
105 (5, 1)
6(5, 2)
7(5, 3)
8(5, 4)
9(5, 5)
10(5, 6)
116 (6, 1)
7(6, 2)
8(6, 3)
9(6, 4)
10(6, 5)
11(6, 6)
12
34) Find the probability of rolling a sum of \(3\).
35) Find the probability of rolling at least one four or a sum of \(8\).
- Answer
-
\(\dfrac{5}{12}\)
36) Find the probability of rolling an odd sum less than \(9\).
37) Find the probability of rolling a sum greater than or equal to \(15\).
- Answer
-
\(0\)
38) Find the probability of rolling a sum less than \(15\).
39) Find the probability of rolling a sum less than \(6\) or greater than \(9\).
- Answer
-
\(\dfrac{4}{9}\)
40) Find the probability of rolling a sum between \(6\) and \(9\), inclusive.
41) Find the probability of rolling a sum of \(5\) or \(6\).
- Answer
-
\(\dfrac{1}{4}\)
42) Find the probability of rolling any sum other than \(5\) or \(6\).
For the exercises 43-46, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:
43) A head on the coin or a club
- Answer
-
\(\dfrac{5}{8}\)
44) A tail on the coin or red ace
45) A head on the coin or a face card
- Answer
-
\(\dfrac{8}{13}\)
46) No aces
For the exercises 47-50, use this scenario: a bag of M&Ms contains \(12\) blue, \(6\) brown, \(10\) orange, \(8\) yellow, \(8\) red, and \(4\) green M&Ms. Reaching into the bag, a person grabs \(5\) M&Ms.
47) What is the probability of getting all blue M&Ms?
- Answer
-
\(\dfrac{C(12,5)}{C(48,5)}=\dfrac{1}{2162}\)
48) What is the probability of getting \(4\) blue M&Ms?
49) What is the probability of getting \(3\) blue M&Ms?
- Answer
-
\(\dfrac{C(12,3)C(36,2)}{C(48,5)}=\dfrac{175}{2162}\)
50) What is the probability of getting no brown M&Ms?
Extensions
Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting \(20\) numbers from the numbers \(1\) to \(80\). After the player makes his selections, \(20\) winning numbers are randomly selected from numbers \(1\) to \(80\). A win occurs if the player has correctly selected \(3,4\) or \(5\) of the \(20\) winning numbers. (Round all answers to the nearest hundredth of a percent.)
51) What is the percent chance that a player selects exactly \(3\) winning numbers?
- Answer
-
\(\dfrac{C(20,3)C(60,17)}{C(80,20)}\approx 12.49\%\)
52) What is the percent chance that a player selects exactly \(4\) winning numbers?
53) What is the percent chance that a player selects all \(5\) winning numbers?
- Answer
-
\(\dfrac{C(20,5)C(60,15)}{C(80,20)}\approx 23.33\%\)
54) What is the percent chance of winning?
55) How much less is a player’s chance of selecting \(3\) winning numbers than the chance of selecting either \(4\) or \(5\) winning numbers?
- Answer
-
\(20.50+23.33-12.49=31.34\%\)
Real-World Applications
Use this data for the exercises 56-60: In 2013, there were roughly \(317\) million citizens in the United States, and about \(40\) million were elderly (aged \(65\) and over)
56) If you meet a U.S. citizen, what is the percent chance that the person is elderly? (Round to the nearest tenth of a percent.)
57) If you meet five U.S. citizens, what is the percent chance that exactly one is elderly? (Round to the nearest tenth of a percent.)
- Answer
-
\(\dfrac{C(40000000,1)C(277000000,4)}{C(317000000,5)}\approx 36.78\%\)
58) If you meet five U.S. citizens, what is the percent chance that three are elderly? (Round to the nearest tenth of a percent.)
59) If you meet five U.S. citizens, what is the percent chance that four are elderly? (Round to the nearest thousandth of a percent.)
- Answer
-
\(\dfrac{C(40000000,4)C(277000000,1)}{C(317000000,5)}\approx 0.11\%\)
60) It is predicted that by 2030, one in five U.S. citizens will be elderly. How much greater will the chances of meeting an elderly person be at that time? What policy changes do you foresee if these statistics hold true?
Contributors and Attributions
-
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .


6.11: The Probability Density Function for the Relative Velocity
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From our development of the Maxwell-Boltzmann probability density functions, we can express the probability that the velocity components of particle 1 lie in the intervals \(v_{1x}\) to \(v_{1x}+dv_{1x}\); \(v_{1y}\) to \(v_{1y}+dv_{1y}\); \(v_{1z}\) to \(v_{1z}+dv_{1z}\); while those of particle 2 simultaneously lie in the intervals \(v_{2x}\) to \(v_{2x}+dv_{2x}\); \(v_{2y}\) to \(v_{2y}+dv_{2y}\); \(v_{2z}\) to \(v_{2z}+dv_{2z}\) as
\[\left(\frac{df\left(v_{1x}\right)}{dv_{1x}}\right)\left(\frac{df\left(v_{1y}\right)}{dv_{1y}}\right)\left(\frac{df\left(v_{1z}\right)}{dv_{1z}}\right)\left(\frac{df\left(v_{2x}\right)}{dv_{2x}}\right)\left(\frac{df\left(v_{2y}\right)}{dv_{2y}}\right)\left(\frac{df\left(v_{2z}\right)}{dv_{2z}}\right) \nonumber \]
\[\times dv_{1x}dv_{1y}dv_{1z}dv_{2x}dv_{2y}dv_{2z} \nonumber \]
\[=\left(\frac{df\left(v_1\right)}{dv_1}\right)\left(\frac{df\left(v_2\right)}{dv_2}\right)dv_1dv_2 \nonumber \]
We want to express this probability using the relative velocity coordinates. Since the velocity of the center of mass and the relative velocity are independent, we might expect that the Jacobian of this transformation is just the product of the two individual Jacobians. This turns out to be the case. The Jacobian of the transformation
\[\left({\dot{x}}_1,{\dot{y}}_1,{\dot{z}}_1,{\dot{x}}_2,{\dot{y}}_2,{\dot{z}}_2\right)\to \left({\dot{x}}_0,{\dot{,y}}_0,{\dot{z}}_0,{\dot{x}}_{12},{\dot{y}}_{12},{\dot{z}}_{12}\right) \nonumber \]
is a six-by-six determinate. It is messy, but straightforward, to show that it is equal to the product of two three-by-three determinants and that the absolute value of this product is one. Therefore, we have
\[\begin{align*} dv_{1x}dv_{1y}dv_{1z}dv_{2x}dv_{2y}dv_{2z} =& d{\dot{x}}_1d{\dot{y}}_1d{\dot{z}}_1d{\dot{x}}_2d{\dot{y}}_2d{\dot{z}}_2 \\[4pt] &=d{\dot{x}}_0d{\dot{y}}_0d{\dot{z}}_0d{\dot{x}}_{12}d{\dot{y}}_{12}d{\dot{z}}_{12} \end{align*} \nonumber \]
We transform the probability density by substituting into the one-dimensional probability density functions. That is,
\[\begin{align*} \left(\frac{df\left(v_1\right)}{dv_1}\right)\left(\frac{df\left(v_2\right)}{dv_2}\right) &={\left(\frac{m_1}{2\pi kT}\right)}^{3/2} \mathrm{exp}\left(\frac{-m_1\left(v^2_{1x}+v^2_{1y}+v^2_{1z}\right)}{2kT}\right)\times {\left(\frac{m_2}{2\pi kT}\right)}^{3/2} \mathrm{exp}\left(\frac{-m_2\left(v^2_{2x}+v^2_{2y}+v^2_{2z}\right)}{2kT}\right) \\[4pt] &={\left(\frac{m_1m_2}{4\pi^2k^2T^2}\right)}^{3/2}\times \mathrm{exp}\left(\frac{-m_1\left(v^2_{1x}+v^2_{1y}+v^2_{1z}\right)-m_2\left(v^2_{2x}+v^2_{2y}+v^2_{2z}\right)}{2kT}\right) \\[4pt] &= \left(\frac{m_1m_2}{4\pi^2k^2T^2}\right)^{3/2} \times \mathrm{exp}\left(\frac{-\frac{m_1m_2}{\mu}\left(\dot{x}^2_0+\dot{y}^2_0+z^2_0\right)-\mu \left(\dot{x}^2_{12} + \dot{y}^2_{12}+\dot{z}^2_{12}\right)} {2kT} \right) \end{align*} \nonumber \]
where the last expression specifies the probability density as a function of the relative velocity coordinates.
Next, we make a further transformation of variables. We convert the velocity of the center of mass, \(\left({\dot{x}}_0,{\dot{y}}_0,{\dot{z}}_0\right)\), and the relative velocity,\(\ \left({\dot{x}}_{12},{\dot{y}}_{12},{\dot{z}}_{12}\right)\), from Cartesian coordinates to spherical coordinates, referred to the \(Oxyz\) axis system. (The motion of the center of mass is most readily visualized in the original frame \(Oxyz\). The relative motion, \({\mathop{v}\limits^{\rightharpoonup}}_{12}\), is most readily visualized in the Particle-One Centered Frame,\(\ O_1x^{''}y^{''}z^{''}\). In \(O_1x^{''}y^{''}z^{''}\), the motion of particle 2 is specified by \({\dot{x}}^{''}_2={\dot{x}}_{12}\), \({\dot{y}}^{''}_2={\dot{y}}_{12}\), and \({\dot{z}}^{''}_2={\dot{z}}_{12}.\) The motion of the center of mass is specified by \({\dot{x}}^{''}_0={\mu {\dot{x}}_{12}}/{m_1}\), \({\dot{y}}^{''}_0={\mu {\dot{y}}_{12}}/{m_1}\), and \({\dot{z}}^{''}_0={\mu {\dot{z}}_{12}}/{m_1}\). Since it is the relative motion that is actually of interest, it might seem that we should refer the spherical coordinates to the \(O_1x^{''}y^{''}z^{''}\) frame. This is an unnecessary distinction because all three coordinate frames are parallel to one another, and \({\mathop{r}\limits^{\rightharpoonup}}_0\) and \({\mathop{r}\limits^{\rightharpoonup}}_{12}\) are the same vectors in all three frames.) Letting
\[v^2_0={\dot{x}}^2_0+{\dot{y}}^2_0+z^2_0 \nonumber \]
\[v^2_{12}={\dot{x}}^2_{12}+{\dot{y}}^2_{12}+z^2_{12} \nonumber \]
the Cartesian velocity components are expressed in spherical coordinates by
\[{\dot{x}}_0=v_0{ \sin \theta_0\ }{\mathrm{cos} {\varphi }_0\ } \nonumber \]
\[{\dot{y}}_0=v_0{ \sin \theta_0\ }{ \sin {\varphi }_0\ } \nonumber \]
\[{\dot{z}}_0=v_0{\mathrm{cos} \theta_0\ } \nonumber \]
\[{\dot{x}}_{12}=v_{12}{ \sin \theta_{12}\ }{\mathrm{cos} {\varphi }_{12}\ } \nonumber \]
\[{\dot{y}}_{12}=v_{12}{ \sin \theta_{12}\ }{ \sin {\varphi }_{12}\ } \nonumber \]
\[{\dot{z}}_{12}=v_{12}{\mathrm{cos} \theta_{12}\ } \nonumber \]
The angles \(\theta_0\), \(\theta_{12}\), \({\varphi }_0\), and \({\varphi }_{12}\) are defined in the usual manner relative to the \(Oxyz\) axis system. The Jacobian of this transformation is a six-by-six determinate; which can again be converted to the product of two three-by-three determinates. We find
\[{d\dot{x}}_0{d\dot{y}}_0{d\dot{z}}_0{d\dot{x}}_{12}{d\dot{y}}_{12}{d\dot{z}}_{12}= \nonumber \] \[=v^2_0{ \sin \theta_0\ }dv_0d\theta_0d{\varphi }_0v^2_{12}{ \sin \theta_{12}\ }dv_{12}d\theta_{12}d{\varphi }_{12} \nonumber \]
The probability that the components of the velocity of the center of mass lie in the intervals \(v_0\) to \(v_0+dv_0\); \(\theta_0\) to \(\theta_0+d\theta_0\); \({\varphi }_0\) to \({\varphi }_0+d{\varphi }_0\); while the components of the relative velocity lie in the intervals \(v_{12}\) to \(v_{12}+dv_{12}\); \(\theta_{12}\) to \(\theta_{12}+d\theta_{12}\); \({\varphi }_{12}\) to \({\varphi }_{12}+d{\varphi }_{12}\); becomes
\[{\left(\frac{m_1m_2}{4\pi^2k^2T^2}\right)}^{3/2}exp\left(\frac{-m_1m_2v^2_0}{2\mu kT}\right)exp\left(\frac{-\mu v^2_{12}}{2kT}\right)\times \nonumber \]
\[v^2_0{ \sin \theta_0\ }dv_0d\theta_0d{\varphi }_0v^2_{12}{ \sin \theta_{12}\ }dv_{12}d\theta_{12}d{\varphi }_{12} \nonumber \]
We are interested in the probability increment for the relative velocityrelative velocity:probability density function irrespective of the velocity of the center of mass. To sum the contributions for all possible motions of the center of mass, we integrate this expression over the possible ranges of \(v_0\), \(\theta_0\), and \({\varphi }_0\). We have
\[\left(\frac{df\left(v_{12}\right)}{dv_{12}}\right)\left(\frac{df\left(\theta_{12}\right)}{d\theta_{12}}\right)\left(\frac{df\left({\varphi }_{12}\right)}{d{\varphi }_{12}}\right)dv_{12}d\theta_{12}d{\varphi }_{12}= \nonumber \]
\[={\left(\frac{m_1m_2}{4\pi^2k^2T^2}\right)}^{3/2}\int^{\infty }_0{v^2_0exp\left(\frac{-m_1m_2v^2_0}{2\mu kT}\right)}dv_0\times \int^\pi_0{ \sin \theta_0\ }d\theta_0\int^{2\pi }_0{d{\varphi }_0}\ \times \left[v^2_{12}exp\left(\frac{-\mu v^2_{12}}{2kT}\right){ \sin \theta_{12}\ }dv_{12}d\theta_{12}d{\varphi }_{12}\right] \nonumber \]
\[={\left(\frac{\mu }{2\pi kT}\right)}^{3/2}v^2_{12}exp\left(\frac{-\mu v^2_{12}}{2kT}\right){ \sin \theta_{12}\ }dv_{12}d\theta_{12}d{\varphi }_{12} \nonumber \]
This is the same as the probability increment for a single-particle velocity—albeit with \(\mu\) replacing \(m\); \(v_{12}\) replacing \(v\); \(\theta_{12}\) replacing \(\theta\); and \({\varphi }_{12}\) replacing \(\varphi\). As in the single-particle case, we can obtain the probability increment for the scalar component of the relative velocity by integrating over all possible values of \(\theta_{12}\) and \({\varphi }_{12}\). We find
\[\frac{df\left(v_{12}\right)}{dv_{12}}=4\pi {\left(\frac{\mu }{2\pi kT}\right)}^{3/2}v^2_{12}exp\left(\frac{-\mu v^2_{12}}{2kT}\right){dv}_{12} \nonumber \]
In § 8, we find the most probable velocity, the mean velocity, and the root-mean-square velocity for a gas whose particles have mass \(m\). By identical arguments, we obtain the most probable relative velocity, the mean relative velocity, and the root-mean-square relative velocity. To do so, we can simply substitute \(\mu\) for \(m\) in the earlier results. In particular, the mean relative velocity is
\[{\overline{v}}_{12}=\left\langle v_{12}\right\rangle ={\left(\frac{8kT}{\pi \mu }\right)}^{1/2}\approx 1.596 {\left(\frac{kT}{\pi \mu }\right)}^{1/21} \nonumber \]
If particles 1 and 2 have the same mass, \(m\), the reduced mass becomes \(\mu ={m}/{2}\). In this case, we have
\[\left\langle v_{12}\right\rangle = {\left(\frac{2\left(8kT\right)}{\pi m}\right)}^{1/2}=\sqrt{2}\left\langle v\right\rangle \nonumber \]
We can arrive at this same conclusion by considering the relative motion of two particles that represents the average case. As illustrated in Figure 9, this occurs when the two particles have the same speed, \(\left\langle v\right\rangle\), but are moving at 90-degree angles to one another. In this situation, the length of the resultant vector—the relative speed— is just
\[\left|{\overline{v}}_{12}\right|=\left\langle v_{12}\right\rangle =\sqrt{2}\left\langle v\right\rangle. \nonumber \]


12.4: The Filibuster and the Quest for New Slave States
Southern expansionists had spearheaded the drive to add more territory to the United States. They applauded the Louisiana Purchase and fervently supported Indian removal, the annexation of Texas, and the Mexican-American War. Drawing inspiration from the annexation of Texas, proslavery expansionists hoped to replicate that feat by bringing Cuba and other territories into the United States and thereby enlarging the American empire of slavery.
In the 1850s, the expansionist drive among white southerners intensified. Among southern imperialists, one way to push for the creation of an American empire of slavery was through the actions of filibusters—men who led unofficial military operations intended to seize land from foreign countries or foment revolution there. These unsanctioned military adventures were not part of the official foreign policy of the United States; American citizens simply formed themselves into private armies to forcefully annex new land without the government’s approval.
An 1818 federal law made it a crime to undertake such adventures, which was an indication of both the reality of efforts at expansion through these illegal expeditions and the government’s effort to create a U.S. foreign policy. Nonetheless, Americans continued to filibuster throughout the nineteenth century. In 1819, an expedition of two hundred Americans invaded Spanish Texas, intent on creating a republic modeled on the United States, only to be driven out by Spanish forces. Using force, taking action, and asserting white supremacy in these militaristic drives were seen by many as an ideal of American male vigor. President Jackson epitomized this military prowess as an officer in the Tennessee militia, where earlier in the century he had played a leading role in ending the Creek War and driving Indian peoples out of Alabama and Georgia. His reputation helped him to win the presidency in 1828 and again in 1832.
Filibustering plots picked up pace in the 1850s as the drive for expansion continued. Slaveholders looked south to the Caribbean, Mexico, and Central America, hoping to add new slave states. Spanish Cuba became the objective of many American slaveholders in the 1850s, as debate over the island dominated the national conversation. Many who urged its annexation believed Cuba had to be made part of the United States to prevent it from going the route of Haiti, with black slaves overthrowing their masters and creating another black republic, a prospect horrifying to many in the United States. Americans also feared that the British, who had an interest in the sugar island, would make the first move and snatch Cuba from the United States. Since Britain had outlawed slavery in its colonies in 1833, blacks on the island of Cuba would then be free.
Narisco López, a Cuban who wanted to end Spanish control of the island, gained American support. He tried five times to take the island, with his last effort occurring in the summer of 1851 when he led an armed group from New Orleans. Thousands came out to cheer his small force as they set off to wrest Cuba from the Spanish. Unfortunately for López and his supporters, however, the effort to take Cuba did not produce the hoped-for spontaneous uprising of the Cuban people. Spanish authorities in Cuba captured and executed López and the American filibusters.
Efforts to take Cuba continued under President Franklin Pierce, who had announced at his inauguration in 1853 his intention to pursue expansion. In 1854, American diplomats met in Ostend, Belgium, to find a way to gain Cuba. They wrote a secret memo, known as the Ostend Manifesto (thought to be penned by James Buchanan, who was elected president two years later), stating that if Spain refused to sell Cuba to the United States, the United States was justified in taking the island as a national security measure.
The contents of this memo were supposed to remain secret, but details were leaked to the public, leading the House of Representatives to demand a copy. Many in the North were outraged over what appeared to be a southern scheme, orchestrated by what they perceived as the Slave Power—a term they used to describe the disproportionate influence that elite slaveholders wielded—to expand slavery. European powers also reacted with anger. Southern annexationists, however, applauded the effort to take Cuba. The Louisiana legislature in 1854 asked the federal government to take decisive action, and John Quitman, a former Mississippi governor, raised money from slaveholders to fund efforts to take the island.
Read an 1860 editorial titled Annexation of Cuba Made Easy from the online archives of The New York Times . Does the author support annexation? Why or why not?
Controversy around the Ostend Manifesto caused President Pierce to step back from the plan to take Cuba. After his election, President Buchanan, despite his earlier expansionist efforts, denounced filibustering as the action of pirates. Filibustering caused an even wider gulf between the North and the South (Figure 12.4.1).
Cuba was not the only territory in slaveholders’ expansionist sights: some focused on Mexico and Central America. In 1855, Tennessee-born William Walker, along with an army of no more than sixty mercenaries, gained control of the Central American nation of Nicaragua. Previously, Walker had launched a successful invasion of Mexico, dubbing his conquered land the Republic of Sonora. In a relatively short period of time, Walker was dislodged from Sonora by Mexican authorities and forced to retreat back to the United States. His conquest of Nicaragua garnered far more attention, catapulting him into national popularity as the heroic embodiment of white supremacy (Figure 12.4.2).
Why Nicaragua? Nicaragua presented a tempting target because it provided a quick route from the Caribbean to the Pacific: Only twelve miles of land stood between the Pacific Ocean, the inland Lake Nicaragua, and the river that drained into the Atlantic. Shipping from the East Coast to the West Coast of the United States had to travel either by land across the continent, south around the entire continent of South America, or through Nicaragua. Previously, American tycoon Cornelius Vanderbilt (Figure 12.4.2) had recognized the strategic importance of Nicaragua and worked with the Nicaraguan government to control shipping there. The filibustering of William Walker may have excited expansionist-minded southerners, but it greatly upset Vanderbilt’s business interests in the region.
Walker clung to the racist, expansionist philosophies of the proslavery South. In 1856, Walker made slavery legal in Nicaragua—it had been illegal there for thirty years—in a move to gain the support of the South. He also reopened the slave trade. In 1856, he was elected president of Nicaragua, but in 1857, he was chased from the country. When he returned to Central America in 1860, he was captured by the British and released to Honduran authorities, who executed him by firing squad.
Section Summary
The decade of the 1850s witnessed various schemes to expand the American empire of slavery. The Ostend Manifesto articulated the right of the United States to forcefully seize Cuba if Spain would not sell it, while filibuster expeditions attempted to annex new slave states without the benefit of governmental approval. Those who pursued the goal of expanding American slavery believed they embodied the true spirit of white racial superiority.
Review Questions
Why did southern expansionists conduct filibuster expeditions?
- to gain political advantage
- to annex new slave states
- to prove they could raise an army
- to map unknown territories
B
The controversy at the heart of the Ostend Manifesto centered on the fate of:
- Ostend, Belgium
- Nicaragua
- Cuba
- Louisiana
C
Why did expansionists set their sights on the annexation of Spanish Cuba?
Many slaveholding expansionists believed that the events of the Haitian Revolution could repeat themselves in Cuba, leading to the overthrow of slavery on the island and the creation of an independent black republic. Americans also feared that the British would seize Cuba—which, since Britain had outlawed slavery in its colonies in 1833, would render all slaves on the island free.
Critical Thinking Questions
Compare and contrast the steamboats of the antebellum years with technologies today. In your estimation, what modern technology compares to steamboats in its transformative power?
Does the history of the cotton kingdom support or undermine the Jeffersonian vision of white farmers on self-sufficient farms? Explain your answer.
Based on your reading of William J. Anderson’s and John Brown’s accounts, what types of traumas did slaves experience? How were the experiences of black women and men similar and different?
What strategies did slaves employ to resist, revolt, and sustain their own independent communities and cultures? How did slaves use white southerners’ own philosophies—paternalism and Christianity, for example—to their advantage in these efforts?
What are the major arguments put forward by proslavery advocates? How would you argue against their statements?
Consider filibustering from the point of view of the Cuban or Nicaraguan people. If you lived in Cuba or Nicaragua, would you support filibustering? Why or why not?
Glossary
- Ostend Manifesto
- the secret diplomatic memo stating that if Spain refused to sell Cuba to the United States, the United States was justified in taking the island as a national security measure


6.7: Drag Force and Terminal Speed
- Express the drag force mathematically
- Describe applications of the drag force
- Define terminal velocity
- Determine an object’s terminal velocity given its mass
Another interesting force in everyday life is the force of drag on an object when it is moving in a fluid (either a gas or a liquid). You feel the drag force when you move your hand through water. You might also feel it if you move your hand during a strong wind. The faster you move your hand, the harder it is to move. You feel a smaller drag force when you tilt your hand so only the side goes through the air—you have decreased the area of your hand that faces the direction of motion.
Drag Forces
Like friction, the drag force always opposes the motion of an object. Unlike simple friction, the drag force is proportional to some function of the velocity of the object in that fluid. This functionality is complicated and depends upon the shape of the object, its size, its velocity, and the fluid it is in. For most large objects such as cyclists, cars, and baseballs not moving too slowly, the magnitude of the drag force \(F_D\) is proportional to the square of the speed of the object. We can write this relationship mathematically as \(F_D \propto v^2\). When taking into account other factors, this relationship becomes
\[F_{D} = \frac{1}{2} C \rho A v^{2}, \label{6.5}\]
where \(C\) is the drag coefficient, \(A\) is the area of the object facing the fluid, and \(\rho\) is the density of the fluid. (Recall that density is mass per unit volume.) This equation can also be written in a more generalized fashion as \(F_D = bv^2\), where b is a constant equivalent to \(0.5C \rho A\). We have set the exponent n for these equations as 2 because when an object is moving at high velocity through air, the magnitude of the drag force is proportional to the square of the speed. As we shall see in Fluid Mechanics , for small particles moving at low speeds in a fluid, the exponent n is equal to 1.
Drag force \(F_D\) is proportional to the square of the speed of the object. Mathematically,
\[F_{D} = \frac{1}{2} C \rho A v^{2},\]
where \(C\) is the drag coefficient, \(A\) is the area of the object facing the fluid, and \(\rho\) is the density of the fluid.
Athletes as well as car designers seek to reduce the drag force to lower their race times (Figure \(\PageIndex{1A}\)). Aerodynamic shaping of an automobile can reduce the drag force and thus increase a car’s gas mileage. The value of the drag coefficient \(C\) is determined empirically, usually with the use of a wind tunnel (Figure \(\PageIndex{1B}\)).
The drag coefficient can depend upon velocity, but we assume that it is a constant here. Table \(\PageIndex{1}\) lists some typical drag coefficients for a variety of objects. Notice that the drag coefficient is a dimensionless quantity. At highway speeds, over 50% of the power of a car is used to overcome air drag. The most fuel-efficient cruising speed is about 70–80 km/h (about 45–50 mi/h). For this reason, during the 1970s oil crisis in the United States, maximum speeds on highways were set at about 90 km/h (55 mi/h).
| Object | C |
|---|---|
| Airfoil | 0.05 |
| Toyota Camry | 0.28 |
| Ford Focus | 0.32 |
| Honda Civic | 0.36 |
| Ferrari Testarossa | 0.37 |
| Dodge Ram Pickup | 0.43 |
| Sphere | 0.45 |
| Hummer H2 SUV | 0.64 |
| Skydiver (feet first) | 0.70 |
| Bicycle | 0.90 |
| Skydiver (horizontal) | 1.0 |
| Circular flat plate | 1.12 |
Substantial research is under way in the sporting world to minimize drag.  Bicycle racers and some swimmers and runners wear full bodysuits. Australian Cathy Freeman wore a full body suit in the 2000 Sydney Olympics and won a gold medal in the 400-m race. Many swimmers in the 2008 Beijing Olympics wore (Speedo) body suits; it might have made a difference in breaking many world records (Figure \(\PageIndex{2}\)). Most elite swimmers (and cyclists) shave their body hair. Such innovations can have the effect of slicing away milliseconds in a race, sometimes making the difference between a gold and a silver medal. One consequence is that careful and precise guidelines must be continuously developed to maintain the integrity of the sport.
Terminal Velocity
Some interesting situations connected to Newton’s second law occur when considering the effects of drag forces upon a moving object. For instance, consider a skydiver falling through air under the influence of gravity. The two forces acting on him are the force of gravity and the drag force (ignoring the small buoyant force). The downward force of gravity remains constant regardless of the velocity at which the person is moving. However, as the person’s velocity increases, the magnitude of the drag force increases until the magnitude of the drag force is equal to the gravitational force, thus producing a net force of zero. A zero net force means that there is no acceleration, as shown by Newton’s second law. At this point, the person’s velocity remains constant and we say that the person has reached his terminal velocity (\(v_T\)). Since \(F_D\) is proportional to the speed squared, a heavier skydiver must go faster for F D to equal his weight. Let’s see how this works out more quantitatively.
At the terminal velocity,
\[F_{net} = mg - F_{D} = ma = 0 \ldotp\]
Thus,
\[mg = F_{D} \ldotp\]
Using the equation for drag force, we have
\[mg = \frac{1}{2} C \rho A v_{T}^{2} \ldotp\]
Solving for the velocity, we obtain
\[v_{T} = \sqrt{\frac{2mg}{\rho CA}} \ldotp\]
Assume the density of air is \(\rho\) = 1.21 kg/m 3 . A 75-kg skydiver descending head first has a cross-sectional area of approximately A = 0.18 m 2 and a drag coefficient of approximately C = 0.70. We find that
\[v_{T} = \sqrt{\frac{2(75\; kg)(9.80\; m/s^{2})}{(1.21\; kg/m^{3})(0.70)(0.18\; m^{2})}} = 98\; m/s = 350\; km/h \ldotp\]
This means a skydiver with a mass of 75 kg achieves a terminal velocity of about 350 km/h while traveling in a pike (head first) position, minimizing the area and his drag. In a spread-eagle position, that terminal velocity may decrease to about 200 km/h as the area increases. This terminal velocity becomes much smaller after the parachute opens.
Find the terminal velocity of an 85-kg skydiver falling in a spread-eagle position.
Strategy
At terminal velocity, \(F_{net} = 0\). Thus, the drag force on the skydiver must equal the force of gravity (the person’s weight). Using the equation of drag force, we find \(mg = \frac{1}{2} \rho C A v^{2}\).
Solution
The terminal velocity \(v_T\) can be written as
\[v_{T} = \sqrt{\frac{2mg}{\rho CA}} = \sqrt{\frac{2(85\; kg)(9.80\; m/s^{2})}{(1.21\; kg/m^{3})(1.0)(0.70\; m^{2})}} = 44\; m/s \ldotp\]
Significance
This result is consistent with the value for v T mentioned earlier. The 75-kg skydiver going feet first had a terminal velocity of v T = 98 m/s. He weighed less but had a smaller frontal area and so a smaller drag due to the air.
Find the terminal velocity of a 50-kg skydiver falling in spread-eagle fashion.
The size of the object that is falling through air presents another interesting application of air drag. If you fall from a 5-m-high branch of a tree, you will likely get hurt—possibly fracturing a bone. However, a small squirrel does this all the time, without getting hurt. You do not reach a terminal velocity in such a short distance, but the squirrel does.
The following interesting quote on animal size and terminal velocity is from a 1928 essay by a British biologist, J. B. S. Haldane, titled “On Being the Right Size.”
“To the mouse and any smaller animal, [gravity] presents practically no dangers. You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, and a horse splashes. For the resistance presented to movement by the air is proportional to the surface of the moving object. Divide an animal’s length, breadth, and height each by ten; its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistance to falling in the case of the small animal is relatively ten times greater than the driving force.”
The above quadratic dependence of air drag upon velocity does not hold if the object is very small, is going very slow, or is in a denser medium than air. Then we find that the drag force is proportional just to the velocity. This relationship is given by Stokes’ law.
For a spherical object falling in a medium, the drag force is
\[F_{s} = 6 \pi r \eta v, \label{6.6}\]
where \(r\) is the radius of the object, \(\eta\) is the viscosity of the fluid, and \(v\) is the object’s velocity.
Good examples of Stokes’ law are provided by microorganisms, pollen, and dust particles. Because each of these objects is so small, we find that many of these objects travel unaided only at a constant (terminal) velocity. Terminal velocities for bacteria (size about \(1\, \mu m) can be about \(2\, \mu m/s. To move at a greater speed, many bacteria swim using flagella (organelles shaped like little tails) that are powered by little motors embedded in the cell.
Sediment in a lake can move at a greater terminal velocity (about 5 \(\mu\)m/s), so it can take days for it to reach the bottom of the lake after being deposited on the surface.
If we compare animals living on land with those in water, you can see how drag has influenced evolution. Fish, dolphins, and even massive whales are streamlined in shape to reduce drag forces. Birds are streamlined and migratory species that fly large distances often have particular features such as long necks. Flocks of birds fly in the shape of a spearhead as the flock forms a streamlined pattern (Figure \(\PageIndex{3}\)). In humans, one important example of streamlining is the shape of sperm, which need to be efficient in their use of energy.
In lecture demonstrations, we do measurements of the drag force on different objects. The objects are placed in a uniform airstream created by a fan. Calculate the Reynolds number and the drag coefficient.
Video \(\PageIndex{1}\) : Fluid Mechanics - Drag force - Flow simulation
The Calculus of Velocity-Dependent Frictional Forces
When a body slides across a surface, the frictional force on it is approximately constant and given by \(\mu_{k}N\). Unfortunately, the frictional force on a body moving through a liquid or a gas does not behave so simply. This drag force is generally a complicated function of the body’s velocity. However, for a body moving in a straight line at moderate speeds through a liquid such as water, the frictional force can often be approximated by
\[f_{R} = -bv,\]
where b is a constant whose value depends on the dimensions and shape of the body and the properties of the liquid, and \(v\) is the velocity of the body. Two situations for which the frictional force can be represented this equation are a motorboat moving through water and a small object falling slowly through a liquid.
Let’s consider the object falling through a liquid. The free-body diagram of this object with the positive direction downward is shown in Figure \(\PageIndex{4}\). Newton’s second law in the vertical direction gives the differential equation
\[mg - bv = m \frac{dv}{dt},\]
where we have written the acceleration as \(\frac{dv}{dt}\). As v increases, the frictional force \(–bv\) increases until it matches mg. At this point, there is no acceleration and the velocity remains constant at the terminal velocity v T . From the previous equation,
\[mg - bv_{T} = 0,\]
so
\[v_{T} = \frac{mg}{b} \ldotp\]
We can find the object’s velocity by integrating the differential equation for \(v\). First, we rearrange terms in this equation to obtain
\[\frac{dv}{g- \left(\dfrac{b}{m}\right)v} = dt \ldotp \label{eq20}\]
Assuming that \(v = 0\) at \9t = 0\), integration of Equation \ref{eq20} yields
\[\int_{0}^{v} \frac{dv'}{g- \left(\dfrac{b}{m}\right)v'} = \int_{0}^{t} dt',\]
or
\[- \frac{m}{b} \ln \left(g - \dfrac{b}{m} v' \right) \Bigg|_{0}^{v} = t' \big|_{0}^{t} ,\]
where \(v'\) and \(t'\) are dummy variables of integration. With the limits given, we find
\[- \frac{m}{b} [ \ln \left(g - \dfrac{b}{m} v \right) - \ln g] = t \ldotp\]
Since \(\ln A − \ln B = \ln (\left(\frac{A}{B}\right)\), and \(\ln (\left(\frac{A}{B}\right) = x\) implies \(e^x = \dfrac{A}{B}\), we obtain
\[\frac{g - \left(\dfrac{bv}{m}\right)}{g} = e^{- \frac{bt}{m}},\]
and
\[v = \frac{mg}{b} \big( 1 - e^{- \frac{bt}{m}} \big) \ldotp\]
Notice that as t → \(\infty\), v → \(\frac{mg}{b}\) = v T , which is the terminal velocity.
The position at any time may be found by integrating the equation for v. With v = \(\frac{dy}{dt}\),
\[dy = \frac{mg}{b} \big( 1 - e^{- \frac{bt}{m}} \big)dt \ldotp\]
Assuming y = 0 when t = 0,
\[\int_{0}^{y} dy' = \frac{mg}{b} \int_{0}^{t} \big( 1 - e^{- \frac{bt}{m}} \big)dt',\]
which integrates to
\[y = \frac{mg}{b} t + \frac{m^{2}g}{b^{2}} \big( e^{- \frac{bt}{m}} - 1 \big) \ldotp\]
A motorboat is moving across a lake at a speed v 0 when its motor suddenly freezes up and stops. The boat then slows down under the frictional force \(f_R = −bv\).
- What are the velocity and position of the boat as functions of time?
- If the boat slows down from 4.0 to 1.0 m/s in 10 s, how far does it travel before stopping?
Solution
- With the motor stopped, the only horizontal force on the boat is f R = −bv, so from Newton’s second law, $$m \frac{dv}{dt} = -bv,$$which we can write as $$\frac{dv}{v} = - \frac{b}{m} dt \ldotp$$Integrating this equation between the time zero when the velocity is v 0 and the time t when the velocity is v, we have $$\int_{0}^{v} \frac{dv'}{v'} = -\frac{b}{m} \int_{0}^{t} dt' \ldotp$$ Thus, $$\ln \frac{v}{v_{0}} = - \frac{b}{m} t,$$which, since lnA = x implies e x = A, we can write this as $$v = v_{0} e^{- \frac{bt}{m}} \ldotp$$Now from the definition of velocity, $$\frac{dx}{dt} = v_{0} e^{- \frac{bt}{m}},$$so we have $$dx = v_{0} e^{- \frac{bt}{m}} dt \ldotp$$With the initial position zero, we have $$\int_{0}^{x} dx' = v_{0} \int_{0}^{t} e^{- \frac{bt'}{m}} dt' ,$$and $$x = - \frac{mv_{0}}{b} e^{- \frac{bt}{m}} \Big|_{0}^{t} = \frac{mv_{0}}{b} \big(1 - e^{- \frac{bt}{m}} \big) \ldotp$$As time increases, \(e^{- \frac{bt}{m}}\) → 0, and the position of the boat approaches a limiting value $$x_{max} = \frac{mv_{0}}{b} \ldotp$$Although this tells us that the boat takes an infinite amount of time to reach x max , the boat effectively stops after a reasonable time. For example, at t = \(\frac{10m}{b}\), we have$$v = v_{0} e^{-10} \simeq 4.5 \times 10^{-5} v_{0},$$whereas we also have $$x = x_{max} \big(1 - e^{-10} \big) \simeq 0.99995x_{max} \ldotp$$Therefore, the boat’s velocity and position have essentially reached their final values.
- With v 0 = 4.0 m/s and v = 1.0 m/s, we have 1.0 m/s = (4.0 m/s) \(e^{(- \frac{bt}{m})(10\; s)}\), so $$\ln 0.25 = - \ln 4.0 = - \frac{b}{m} (10\;s),$$and $$\frac{b}{m} = \frac{1}{10} \ln 4.0\; s^{-1} = 0.14\; s^{-1} \ldotp$$Now the boat's limiting position is $$x_{max} = \frac{mv_{0}}{b} = \frac{4.0\; m/s}{0.14\; s^{-1}} = 29\; m \ldotp$$
Significance
In the both of the previous examples, we found “limiting” values. The terminal velocity is the same as the limiting velocity, which is the velocity of the falling object after a (relatively) long time has passed. Similarly, the limiting distance of the boat is the distance the boat will travel after a long amount of time has passed. Due to the properties of exponential decay, the time involved to reach either of these values is actually not too long (certainly not an infinite amount of time!) but they are quickly found by taking the limit to infinity.
Suppose the resistive force of the air on a skydiver can be approximated by \(f = −bv^2\). If the terminal velocity of a 100-kg skydiver is 60 m/s, what is the value of b?


41.3.22: Chapter 23
Review Questions
1.
b.
The symptoms presented are classic signs of rhinitis.
2.
a.
Asthma is characterized as difficulty moving air into and out of the lungs due to bronchoconstriction. This causes wheezing during inspiration and expiration.
3.
d.
Chronic bronchitis is characterized by chronic cough and sputum production. The most common cause is smoking.
4.
a.
Identifying triggers will help the client decrease symptoms by avoiding the trigger items.
5.
d.
Emphysema develops gradually, and many clients have no symptoms at first. Most clients notice shortness of breath when doing normal activities.
6.
a.
Clients with COPD are more prone to respiratory infections, so they should implement infection prevention strategies.
7.
b.
Oxygen saturation is the amount of oxygen bound to hemoglobin.
8.
a.
Emphysema and COPD are most often caused by cigarette smoking.
9.
b.
Quitting smoking will not cure the disease, but it can prevent more damage.
10.
c.
Sinus pain and pressure are common symptoms of sinusitis. Using pain reducers such as acetaminophen or ibuprofen can help with these symptoms.


1.S: What is Art? (Summary)
Key Concepts
When studying a subject, it is important to have a working definition of that subject. Our subject is art. The four historical attempts at defining art surveyed here each had limitations. Ancient Greek mimesis excluded art that does not re-present objects. Tolstoy’s communication theory is unverifiable and is spectator-dependent, Bell’s significant form is circular reasoning, and Dickie’s Artworld theory is about who has the power to decide what art is, not about art itself. The operating definition of art used in this text is “from the mind into the world.” The images used in this survey are considered works of art. It is the task of the student to be able to recognize, analyze, and interpret works of art, and to integrate this understanding into a coherent worldview. The purpose of this effort at understanding is to practice recognizing value in new and diverse forms of visual art. One end result is to then have a greater appreciation of and to simply enjoy looking at art.
Art is found wherever we find human beings. Art fulfills a basic human need for expression. This need can be sub-divided into personal needs and needs of the community. Personal needs include art created for delight, decoration, for political and religious devotion, and for personal catharsis. Communal needs can include architecture, monuments, murals, and religious and secular icons.
Glossary
Architecture: the design and construction of buildings or other complex structures.
Artworld theory of art: an approach to defining art as whatever the artworld says it is.
Catharsis: the process of releasing pent up emotion resulting in personal change.
Circumambulate: to “walk around”—a ritual practice of circling a sacred site, following a set path either inside or outside of a structure.
Communication theory of art: an approach to defining art as a transfer of feeling from artist to spectator.
Convention: group consensus about the way something is usually done.
Icon: a person or thing regarded as representative of something, often religious.
Institutional theory of art: another name for the Artworld theory of art.
Labyrinth: similar to a maze, but generally has only one intricate and twisting path to the center.
Mimesis: an approach to defining art as a copy of perceived reality.
Monument: a statue or other structure meant to commemorate a famous person or event.
Mural: a work of art executed directly on a wall.
Relic: an object thought to have belonged to or been part of a holy person’s body.
Secular: lacking in religious or spiritual content, not bound by religious rule.
Significant Form: an approach to defining art as what we notice.
Symbolism: the use of images to represent ideas or qualities.
Trompe l’oeil: art so realistic that it “fools the eye.”
Zeuxis and Parhassios: an ancient Greek myth about two competing painters who vie for the title of greatest artist by copying reality most faithfully
Test Yourself
1. List and describe the four ways stated in the text in which people have defined art in the past.
2. Briefly re-state the operating definition of art for this text.
3. What is the significance of the ancient Greek myth of Zeuxis and Parhassios?
4. What do each of the four historical definitions of art reveal of how people thought about where truth is to be found?
5. Draw parallels between the sea snail shell necklace of c. 100,000 BCE and modern practices of personal decoration, for example, a pearl necklace.
6. Speculate about why images might be important in non-literate cultures? What might be one concern about images used in religious rituals? Can you identify an example of a non-religious icon other than the one noted in the text?
7. Speculate about why most early American federal buildings were built using classical Greek and Roman columns and imposing stone facades. Why were buildings in the twentieth century built with little reference to the architecture of classical antiquity? What ideas were lost and what ideas were gained with this shift in architecture?
8. Consider the change in the conventional presentation of public monuments by comparing how the monuments of Verrocchio and Rodin are presented, one on a high pedestal, the other at ground level. What does this change suggest about changing ideas about the heroic and monumental?


1.5: A Word About Values
Three types of values
Values are standards or ideals with which we evaluate actions, people, things, or situations. Beauty, honesty, justice, peace, generosity are all examples of values that many people endorse. In thinking about values it is useful to distinguish them into three types:
- Personal values: values endorsed by an individual. For example, some people regard family as their most important value and structure their lives so that they can spend more time with their family. Other people might value success instead, and give less time to their families in order to achieve their goals.
- Moral values: values that help determine what is morally right or wrong, e.g. freedom, fairness, equality, well-being, etc… Those which are used to evaluate social institutions are sometimes also known as political values.
- Aesthetic values: values associated with the evaluation of artwork or beauty.
The Role of Facts and Values
The following are descriptive statements. they purport to describe facts :
- This is a sharp knife.
- Mozart and Beethoven are composers.
- Ann believes that freedom of speech is important.
These, however, are statements about values :
- This knife is a very useful kitchen tool
- Mozart is a greater composer than Beethoven
- Freedom of speech should be protected
Philosophers usually distinguish between two kinds of values: intrinsic and extrinsic. Something is supposed to have extrinsic value when it is not valued for its own sake, but because it contributes to some further purpose, or because it helps bring about something else of value. So a particular kitchen knife might be said to be very valuable in this extrinsic sense – it is valued not for its own sake but because it can be used to satisfy certain culinary purposes that we treasure.