The final type of GRE formula problem involves sequences. A sequence is a collection of numbers in a set order. The order of a given sequence is determined by a rule. Here are examples of sequence rules:
For all integers n ≥ 1…
| An = 9n + 3 | The nth term of this sequence is defined by the rule 9n + 3, for integers n ≥ 1. For example, the fourth term in this sequence is 9n + 3 = 9(4) + 3 = 39. The first 10 terms of the sequence are as follows: 12, 21, 30, 39, 48, 57, 66, 75, 84, 93 (notice that successive terms differ by 9) |
| Qn = n2 + 4 | The nth term of this sequence is defined by the rule n2 + 4, for integers n ≥ 1. For example, the first term in this sequence is 12 + 4 = 5. The first 10 terms of the sequence are as follows: 5, 8, 13, 20, 29, 40, 53, 68, 85, 104 |
In these cases, each item of the sequence is defined as a function of n, the place in which the term occurs in the sequence. For example, the value of A5 is a function of its being the fifth item in the sequence. This is a direct definition of a sequence formula.
The GRE also uses recursive formulas to define sequences. With direct formulas, the value of each item in a sequence is defined in terms of its item number in the sequence. With recursive formulas, each item of a sequence is defined in terms of the value of previous items in the sequence.
A recursive formula looks like this:
An = An−1 + 9
This formula simply means “This term (An) equals the previous term (An−1) plus 9.” It is shorthand for a series of specific relationships between successive terms:
Whenever you look at a recursive formula, articulate its meaning in words in your mind. If necessary, also write out one or two specific relationships that the recursive formula stands for. Think of a recursive formula as a “domino” relationship: if you know A1, then you can find A2, and then you can find A3, then A4, and so on for all the terms. You can also work backward: If you know A4, then you can find A3, A2, and A1. However, if you do not know the value of any one term, then you cannot calculate the value of any other. You need one domino to fall, so to speak, to knock down all the others.
Thus, to solve for the values of a recursive sequence, you need to be given the recursive rule and also the value of one of the items in the sequence. For example:
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An = An−1 + 9 |
In this example, An is defined in terms of the previous item, An−1. Recall the meaning of this recursive formula: This term equals the previous term plus 9. Because A1 = 12, you can determine that A2 = A1 + 9 = 12 + 9 = 21. Therefore, A3 = 21 + 9 = 30, A4 = 30 + 9 = 39, and so on. |
Because the first term is 12, this sequence is identical to the sequence defined by the direct definition, An = 9n + 3, given at the beginning of this section. Here is another example:
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Fn = Fn−1 + Fn−2 |
In this example, Fn is defined in terms of both the previous item, Fn−1, and the item prior to that, Fn−2. This recursive formula means “This term equals the previous term plus the term before that.” Because F1 = 1 and F2 = 1, you can determine that F3 = F1 + F2 = 1 + 1 = 2. Therefore, F4 = 2 + 1 = 3, F5 = 3 + 2 = 5, F6 = 5 + 3 = 8, and so on. There is no simple direct rule for this sequence. |
Sn = 2n − 5 for all integers n ≥ 1. What is the 11th term of the sequence?
Bn = (−1)n × n + 3 for all integers n ≥ 1. What is the 9th term of the sequence?
If An = 2An−1 + 3 for all n ≥ 1, and A4 = 45, what is A1?