In the sequence g_n defined for all positive… | GMAT Sprint

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PS · Problem Solving

705-805

Functions

In the sequence $g_n$ defined for all positive integer values of n, $g_1 = g_2 = 1$ and, for n ≥ 3, $g_n = g_{n–1} + 2^{n-3}$ . If the function $ψ(g_i)$ equals the sum of the terms $g_1$ , $g_2$ , …, $g_i$ , what is $\frac{ψ(g_{16})}{ψ(g_{15})}$

Correct answer marked below

g_3

Correct

g_8

ψ(g_8)

ψ(g_{16}) - ψ(g_{15})

\frac{g_{16}}{2}

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Explanation

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Official Explanation

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Given:

g_1 = g_2 = 1

and, for n ≥ 3,

g_n = g_{n–1} + 2^{n-3}

We can compute a few terms:

g_3 = g_2 + 2^{3-3} = 1 + 1 = 2 = g_1+g_2

g_4 = g_3 + 2^{4-3} = 2 + 2 = 4 = g_1+g_2+g_3

g_5 = g_4 + 2^{5-3} = 4 + 4 = 8 = g_1+g_2+g_3+g_4

g_6 = g_5 + 2^{6-3} = 8 + 8 = 16 = g_1+g_2+g_3+g_4+g_5

Observing the pattern:

g_n = g_1+g_2+...+g_{n-1}

for n ≥ 3.

Thus,

g_{15} = g_1+g_2+...+g_{14}

and

g_{16} = g_1+g_2+...+g_{15} = 2 \cdot g_{15}

.…

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What this question tests

Your ability to evaluate and compose functions, including unusual defined operations.

Applying function operations in the wrong order or ignoring domain restrictions.