Let $f : N \rightarrow R$ be a function such that $f( x + y ) = 2 f(x) f(y)$ for natural numbers $x$ and $y$. If $f(1) = 2$, then the value of $\alpha$ for which $\sum _ { k = 1 } ^ { 10 } f ( \alpha + k ) = \frac { 512 } { 3 } ( 2 ^ { 20 } - 1 )$ holds, is\\
(1) 3\\
(2) 4\\
(3) 5\\
(4) 6