For $k \ge 3$, $P(k, n)$ denotes the number of dots that create a regular $k$-gon with side length $n$. For example, here are the few first triangular numbers $P(3, n)$ and the first few square numbers $P(4, n)$ respectively: Here are the first few pentagonal numbers $P(5, n)$: (a) Find all arithmetic sequences of the form $$(P(k, n), P(k, n + 1), P(k + 1, n + 1))$$ (b) For $a \in Z$, show that there are infinitely many arithmetic sequences of the form $$(P(k, n), P(k, n + a), P(k + a, n + a))$$ Note: $n$ is a positive integer here, a member of the set $\{1, 2, 3,...\}$.