$\text{ If }\mathrm{f}:\mathrm{N}\to \mathrm{R}\text{ be a function such that }\mathrm{f}(\mathrm{x}+\mathrm{y})=2\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right)\text{ for natural numbers }\mathrm{x}\text{ and y. If }$$\mathrm{f}\left(1\right)=2\text{, then the value of }\mathrm{\alpha }\text{ for which}\sum _{k=1}^{10} \mathrm{f}(\mathrm{\alpha }+\mathrm{k})=\frac{512}{3}\left(2^{20}-1\right)\text{ holds, is }$

$$\begin{gathered} \begin{array}{l}\mathrm{f}:\mathrm{N}\to \mathrm{R},\mathrm{f}(\mathrm{x}+\mathrm{y})=2\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right)\dots \dots \dots ..\left(1\right) \\ \mathrm{f}\left(1\right)=2, \\ \sum _{\mathrm{k}=1}^{10} \mathrm{f}(\mathrm{\alpha }+\mathrm{k})=2\mathrm{f}\left(\mathrm{\alpha }\right)\sum _{\mathrm{k}=1}^{10} \mathrm{f}\left(\mathrm{k}\right)\\ =2\mathrm{f}\left(\mathrm{\alpha }\right)\left(\mathrm{f}\right(1)+\mathrm{f}(2)+\dots \dots +\mathrm{f}(10\left)\right)\dots \dots \dots ..\left(2\right) \\ \text{ From }\left(1\right)\mathrm{f}\left(2\right)=2{\mathrm{f}}^2\left(1\right)=2^3\\ \mathrm{f}\left(3\right)=2\mathrm{f}\left(2\right)\mathrm{f}\left(1\right)=2^5 \\ \mathrm{f}\left(10\right)=2^9{\mathrm{f}}^{10}\left(1\right)=2^{19} \\ \mathrm{f}\left(\mathrm{\alpha }\right)=2^{2\mathrm{\alpha }-1};\mathrm{\alpha }\in \mathrm{N}\end{array} \end{gathered}$$
From $\text{ (2) }\sum _{\mathrm{k}=1}^{10} \mathrm{f}(\mathrm{\alpha }+\mathrm{k})=2\left(2^{2\mathrm{\alpha }-1}\right)\left(2+2^3+2^5+\dots ..+2^{19}\right)$
$$\begin{gathered} \Rightarrow \frac{512}{3}\left(2^{20}-1\right)=2^{2\mathrm{\alpha }}\left(2\frac{\left(2^{20}-1\right)}{3}\right) \\ \text{Hence }\mathrm{\alpha }=4 \end{gathered}$$