cannot find intuition to solve problem.
Let $F_n = \sum_{j=1}^{n} X_j , j \geq 1$ where $X_j = 1$ if a coin toss is heads and $X_j=-1$ if a coin toss is tails.
I first wanted to show it was a martingale, and I did so by showing $F_n = E_n[F_{n+1}] = 0$.
The next part is where it gets a bit tricky. I am asked to show that for: $L_n = \sum_{j=0}^{n-1} F_j (F_{j+1} - F_j) = \frac{F_j^2 - n}{2} $, for $ n \geq 1$
I understand the logic. $L_n$ describes the cumulative sum of $M_j$ multiplied by the difference between $M_j$ and the expectation of $M_{j+1}$. So, in a sense I see 2 cases, and the $\frac{F_j^2 - n}{2}$ is the 'average' of the two.
However, I cannot rearrange the equation to get the required result. I have tried properties of conditional expectation and martingale, but cannot transform the equation correctly.
Any insight or clues would be much appreciated. Perhaps I am missing some properties of martingales, expectation, or probability. However, my book does not delve far into the matters.