I am trying to solve the following:
Let $\{a_n\}_{n\in \Bbb N}$ be a decreasing sequence of positive terms. Check that:
$\frac{1}{2}(a_1+2a_2+4a_4+...+2^na_{2^n})\le s_{2^n}$ with $s_{2^n}$ $=\sum_{k=0}^n a_{2^k}$
I was thinking of showing that: $$2\times \sum_{k=0}^n a_{k}\ge\sum_{k=0}^n 2^ka_{k}\implies 2\times \sum_{k=0}^n a_{2^k}\ge\sum_{k=0}^n 2^ka_{2^k}$$is this valid?