Mathematics - Sequence & Series Question with Solution | TestHub

$f(x)=2^x$ for all $x\in N$

$$\begin{array}{rl}& \therefore \sum _{k=1}^{n}f(a+k)=16\left(2^n-1\right)\\ & \Rightarrow \sum _{K=1}^n{2}^{a+k}=16\left(2^n-1\right)\\ & \Rightarrow \sum _{k=1}^n{2}^a\cdot 2^k=16\left(2^n-1\right)\\ & \Rightarrow 2^a\left(\sum _{k=1}^n{2}^k\right)=16\left(2^n-1\right)\\ & \Rightarrow 2^a\left(2+2^2+\dots .+2^n\right)=16\left(2^n-1\right)\\ & \Rightarrow 2^a\left\{2\left(\frac{2^n-1}{2-1}\right)\right\}=16\left(2^n-1\right)\\ & \Rightarrow 2^{a+1}\left(2^n-1\right)=16\left(2^n-1\right)\\ & \Rightarrow 2^{a+1}=2^4\\ & \Rightarrow a+1=4\Rightarrow a=3\end{array}$$