If n is a positive integer greater than 2 and f(n)= (1+ 5)^n2^n, what is f(n+1)-f(n-1) in terms of f(n)?

If n is a positive integer greater than 2 and… | GMAT Sprint

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PS · Problem Solving

705-805

Functions

If n is a positive integer greater than 2 and $f(n)=\frac{(1+\sqrt{5})^n}{2^n}$ , what is $f(n+1)-f(n-1)$ in terms of $f(n)$ ?

Answer Choices

Correct answer marked below

\frac{f(n)}{2}

\sqrt{f(n)}

f(n)

Correct

2f(n)

(f(n))^2

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Explanation

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Official Explanation

Official

First, simplify the function using exponent rules:

f(n) = \frac{(1+\sqrt{5})^n}{2^n} = \left(\frac{1+\sqrt{5}}{2}\right)^n

Let

X = \frac{1+\sqrt{5}}{2}

, so

f(n) = X^n

.

The problem asks for

f(n+1) - f(n-1) = X^{n+1} - X^{n-1}

in terms of

f(n)

.

Factor out

X^n

(which is

f(n)

X^{n+1} - X^{n-1} = X^n \cdot (X^1 - X^{-1}) = f(n) \cdot \left( X - \frac{1}{X} \right)

Substitute

X = \frac{1+\sqrt{5}}{2}

f(n) \cdot \left( \frac{1+\sqrt{5}}{2} - \frac{2}{1+\sqrt{5}} \right)

…

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What this question tests

What it tests

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

Common trap

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

If n is a positive integer greater than 2 and f(n)=(1+5)n2nf(n)=\frac{(1+\sqrt{5})^n}{2^n}f(n)=2n(1+5​)n​, what is f(n+1)−f(n−1)f(n+1)-f(n-1)f(n+1)−f(n−1) in terms of f(n)f(n)f(n)?