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

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

by Ian Stewart » Sat Jun 05, 2021 8:59 am
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
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