Here's the problem I'm working with, what I'm stumped on is below:

A biased coin has probability p of coming up Heads. The coin is tossed repeatedly. Let N be the number of tosses until Heads appears for the first time. By conditioning on the outcome of the first toss, find the expected values of

and

. Use this to find the variance of

.

So, I know how to compute

&

here without conditioning on the first toss (i.e. I know what the answers are), but I'm a little stumped on how to get

via that route. I know I can utilize

, but I'm not sure how to solve

. I think a little nudge in the right direction is all I need! Thanks