Understand how to handle these tricky upper level Quant problems!
A sequence is a list of numbers that follow some mathematical patterns. More formally, a sequence is a function whose inputs are limited to the positive integers. Terms are denoted by a letter
Understand how to handle these tricky upper level Quant problems!
A sequence is a list of numbers that follow some mathematical patterns. More formally, a sequence is a function whose inputs are limited to the positive integers. Terms are denoted by a letter for the whole sequence, and in the subscript, the index, which is the place on the list. Here are some examples of common sequences. First, (A) and (B) are arithmetic sequences:
A),,,,, …
B),,,,, …
In an arithmetic sequence, the same “common difference” is added or subtracted each time. In (A), we add 3 to each term to get the next. In (B), we subtract 4 to get each new term. If you graphed the terms of an arithmetic sequence against the indices, the dots would follow a straight line in the x-y plane.
The next two, sequences (C) and (D), are geometric sequences:
C),,,,, …
D),,,,, …
In a geometric sequence, the same “common ratio” is multiplied or divided each time. In (C), we multiply each term by 3 to get the next. In (D), we divide by 2 to get each new term. If you graphed the terms of an geometric sequence against the indices, the dots would follow an exponential function in the x-y plane.
Notice also — (D) begins with a “zeroth” term. The first term of a sequence can have an index of either 1 or 0: both appear on the GMAT.
There are other exotic sequences in mathematics. Here are two more
E),,,,,,,, …
F),,,,,,,, …
Both of these are famous in mathematics. The first is the Fibonacci sequence, which I will discuss further below. The second, known in math as the Partition Function (http://en.wikipedia.org/wiki/Partition_(number_theory)) gets into way way more difficult math that you need to know for the GMAT.
A explicit series is a series in which the general rule for finding each term can be stated, either verbally or mathematically. Sometimes the GMAT will give you the general rule for a sequence in algebraic form:
If the rule is given algebraically, all we have to do is plug in the index number to find the value of each term. Here, we plug in n = 3 to find the third term, and plug in n = 4 to find the fourth term. The third and fourth terms are:
so the difference between them is 12 – 6 = 6, which is answer B. I will give another practice question of this genre at the end.
The explicit description might also be given verbal. For example
2) In each term of a sequence, 9 is added to get the next term. If the first term is 2, what is the eighty-first term?
Notice, the rule for the whole sequence is articulated: we know it is an arithmetic sequence with a common difference of 9. You can try this question now. I will show the solution at the end of this article.
In mathematics, a recursive definition is one where you need the result at end of each step in order to proceed to the next step. A sequence has a recursive definition if the only way to calculate the third term is first to calculate the second term, and the only way to calculate the fourth term is first to calculate the third term, etc. etc. Clearly, if we were asked for, say, the sixth term of a recursive sequence, we would have to calculate each and every term along the way: there’s no “shortcut” we can use to shoot directly to the sixth term.
Recursive defined sequence on the GMAT are almost always given in algebraic form. Here’s an example of a question involving one.
Whenever you see the nth term equal to some expression involving an (n – 1)th term, that’s a recursively defined sequence. The idea is: to find the n = 3 term, you would have to plug the n – 1 = 2 term into the formula; to find the n = 4 term, you would have to plug the n – 1 = 3 term into the formula; etc. I will discuss a full solution to this one below.
Sometimes, a sequence is recursively defined not simply in terms of the previous term, but in terms of the previous two 