Series Solutions · Superchallenge problem

Let $y = \sum_{n=0}^{\infty} a_n x^n$ be the power-series solution of $(1 - 4x)\,y' = 2y$ with $y(0) = 1$. Find the exact value of $\displaystyle\sum_{n=0}^{\infty} \dfrac{a_n}{(n+1)\,8^n}$.