For a geometric random variable Y, the probability distribution is:- $$ P(Y) = q^{(y-1)} p $$
Using $$ E(Y) = \sum_{y=1}^{\infty}yq^{y-1}p $$ with $$ p = 1 - q$$
--
$$ E[Y(Y-1)] = \sum_{y=1}^{\infty}y(y-1)q^{y-1}p$$ $$ = \sum_{y=0}^{\infty}y(y+1)q^{y}p $$
How is this change in equation possible with just adding one more term? I do not understand the steps in between that lead to this transformation. How could those two equations be equivalent?
I understand that for y = 0, the first term would result in 0.