To find the number of possible functions with a given property How many $f$ are possible having this property:
$f:\{1,2,....n\} \rightarrow \mathbb R$ such that
$\big(\sum_{i=1}^{10}\dfrac{|f(i)|}{2^i})^2=(\sum _{i=1}^{10}|f(i)|^2)(\sum _{i=1}^{10}\dfrac{1}{4^i})$
One such $f$ would be $f\equiv 0$.But how to find number of such possible $f$ .Whether it is countable/uncountable?

Let $k$ be any constant, and let $f(i)=\frac{k}{2^i}$. That gives us $c$ solutions, where $c$ is the cardinality of the continuum. And it is clear that there can be no more than $c$ solutions. 
