Question

Compute
\[\log_2(\sum_{k=0}^{1024} C(1024,k) *k)\]

Answer 1

Big thanks to @experimentX
\[(1+x)^n = \sum_{k=0}^{n}\left(\begin{matrix}n \\ k\end{matrix}\right)x^k\]\[n(1+x)^{n-1} = \sum_{k=0}^{n}\left(\begin{matrix}n \\ k\end{matrix}\right)kx^{k-1}\]x=1, n=1024
\[1024(1+1)^{1024-1} = \sum_{k=0}^{1024}\left(\begin{matrix}1024 \\ k\end{matrix}\right)k(1)^{k-1}\]\[\sum_{k=0}^{1024}\left(\begin{matrix}1024 \\ k\end{matrix}\right)k=1024(1+1)^{1023} =...\]