CAT 2019 Functions — Previous Year Question

Consider a function $f(x+y) = f(x) \times f(y)$ where $x$, $y$ are positive integers, and $f(1) = 2$. If $f(a+1) + f(a+2) + \text{...} + f(a+n) = 16 (2^n - 1)$ then $a$ is equal to.