Problem

A progression has first term $a$ and second term $\frac{a^2}{a + 2}$, where $a$ is a positive constant.

(a) For the case where the progression is geometric and the sum to infinity is $264$, find the value of $a$.

(b) For the case where the progression is arithmetic and $a = 6$, determine the least value of $n$ required for the sum of the first $n$ terms to be less than $-480$.