Expected Value

Stack has a positive integer $P(P > 1)$. He starts a game where his initial score is $0$. In the $i^{th}$ turn, with a probability of $\frac{1}{2}$, Stack adds $\frac{1}{P^{i-1}}$ to his score. Stack stops after $N$ turns. Let $S_i$ be his score after $i$ turns and $F(i)$ denote the [expected value](https://en.wikipedia.org/wiki/Expected_value) of $S_i^2$. For each integer $i(1 \leq i \le