Consider a function f satisfying $f(x + y) = f(x)f(y)$ where x, y are positive integers, and $f(1) = 2$ . If $f(a + 1) + f(a + 2) + … + f(a + n) = 16(2^n − 1)$ then $a$ is equal to

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f(x + y) = f(x)f(y)

i.e.

f(a + 1) = f(a)f(1) = 2f(a)

f(a + 2) = f[(a +1)+ 1] = f(a+1)f(1) = 4f(a) = 2^2f(a)

f(a + 3) = f[(a +2)+ 1] = f(a+2)f(1) = 8f(a) = 2^3f(a)

and so on

f(a + 1) + f(a + 2) + … + f(a + n) = 2f(a)[1+2+2^2+2^3+...+2^{n-1}] = 16(2^n − 1)

Sum of n terms of a Geometric Progression:

a+ar+ar^2+ar^3+...+ar^{n-1} = \frac{a(r^n-1)}{(r-1)}

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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.