Suppose that for $1 \leq k < N$, $p_k = \dfrac{1}{2}(p_{k+1} + p_{k-1})$.
Let $b_k = p_k - p_{k-1}$. I am trying to prove that $b_k = b_{k-1}$. I've tried rearranging the terms in $p_k - p_{k-1}$ but was not able to get anywhere.
I would appreciate a small hint to point me in the right direction, thanks!
$$p_k = \frac12 (p_{k+1} + p_{k-1})$$
$$\frac12p_k + \frac12 p_k = \frac12 (p_{k+1} + p_{k-1})$$
Get rid of $\frac12$ and try to express things in terms of $b$.