What is the difference between the fourth and third terms of the sequence defined by \(a_n=3^n-n^2?\)

[BTGmoderatorLU]

Source: Magoosh

What is the difference between the fourth and third terms of the sequence defined by \(a_n=3^n-n^2?\)

A. 18
B. 23
C. 47
D. 65
E. 83

The OA is C

[Ian Stewart]

What is the first term of the sequence? The question needs to tell you - some sequences start at a_0, and some at a_1 (both in real math and on the GMAT), so if the question doesn't specify where the sequence starts, there's no way to know if the "third term" is a_2 or a_3. The GMAT would always be clear about that point.

I'll assume they mean to ask for a_4 - a_3 here. Then plugging into the sequence definition, we find:

a_4 = 3^4 - 4^2

and

a_3 = 3^3 - 3^2

so

a_4 - a_3 = 3^4 - 4^2 - (3^3 - 3^2) = 3^4 - 4^2 - 3^3 + 3^2 = 3^2(3^2 - 3 + 1) - 16 = (9)(7) - 16 = 47