In the sequence g_n defined for all positive integer

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 \(\frac{ψ(g_{16})}{ψ(g_{15})}\)

(A) \(g_3\)

(B) \(g_8\)

(C) \(ψ(g_8)\)

(D) \(ψ(g_{16}) - ψ(g_{15})\)

(E) \(\frac{g_{16}}{2}\)


MANHATTAN GMAT OFFICIAL SOLUTION:

We begin by listing some values of gn, in order to get a sense for how gn progresses:
\(g_1 = 1\)

\(g_2 = 1\)

\(g_3 = g_2 + 2^0 = 1 + 1 = 2 = 2^1\)

\(g_4 = g_3 + 2^1 = 2 + 2 = 4 = 2^2\)

\(g_5 = g_4 + 2^2 = 4 + 4 = 8 = 2^3\)

\(g_6 = g_5 + 2^3 = 8 + 8 = 16 = 2^4\)

We can see that for n ≥ 3, \(g_n = 2^{n–2}\).

Let us now look for a pattern in the sums defined as (gn):
\(ψ(g_3) = g_1 + g_2 + g_3 + = 1 + 1 + 2 = 4 = 2^2\)

\(ψ(g_4) = (g_1 + g_2 + g_3) + g_4 = ψ(g_3) + g_4 = 4 + 4 = 8 = 2^3\)

\(ψ(g_5) = (g_1 + g_2 + g_3 + g_4) + g_5 = ψ(g_4)+ g_5 = 8 + 8 = 16 = 2^4\)

Each value is double the previous value: \(ψ(g_n) = 2 * ψ(g_{n-1})\). This means that:
\(\frac{ψ(g_{16})}{ψ(g_{15})}=\frac{2*ψ(g_{15})}{ψ(g_{15})}=2\)

Now all we need to do is scan the answer choices to find an expression that equals 2. We have already discovered that g3 = 2, so we can select g3 as the answer.

The correct answer is A.

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-the-sequence-gn-defined-for-all-positive-integer-values-of-n-g1-199690.html