Problem Solving

zagcollins wrote:The infinite sequence a1, a2,…, an,… is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an = an-4 for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80

If an = an-4 for all n greater than 4, then a5 = a1, a6 = a2, a7 = a3, a8 = a4, a9 = a5 = a1, etc. So the sequence is just:

2, -3, 5, -1, 2, -3, 5, -1, 2, -3, 5, -1 ....

This repeats in blocks of four. The sum of the first four terms is 3. If we add the first 96 terms, we'll just be adding that same block of four terms 24 times; the sum will be 24*3 = 72. If we add on the 97th term, which will be 2, we get the final answer: 72 + 2 = 74.