f is a function defined as $\sum _{k=1}^{n}f(a+k)=16\left(2^n-1\right)$ and $f(x+y)=f\left(x\right).f\left(y\right)$ and f(1) = 2 then integral value of a

 

$$\begin{gathered} f\left(x+y\right)=f\left(x\right)f\left(y\right) \\ x=y=1\Rightarrow f\left(2\right)=f\left(1\right)f\left(1\right)=4 \\ x=1,y=2\Rightarrow f\left(3\right)=f\left(1\right)f\left(2\right)=8\Rightarrow f\left(x\right)=2^x \\ \sum _{k=1 }^n{2}^{a+k}=16(2^n-1) \\ {2}^a(2+2^2+2^3+...........+2^n)=16(2^n-1) use G.P formula \end{gathered}$$