Determine distribution, Prove Laplace Transform

Consider a game in which a fair coin is tossed indefinitely. Every time heads appears you move 1 metre to the right, and if tails you stay where you are. You start at position 0. Let $\displaystyle T_n $ be the number of tosses needed to first enter position n.

By definition $\displaystyle T_0 = 0 $

Determine the distribution of $\displaystyle T_1 $ and prove that the Laplace Transform of $\displaystyle T_n $ is $\displaystyle \mathbb{E} e^{-sT_n} = (2e^s - 1)^{-n}, s \geq 0 $

Re: Determine distribution, Prove Laplace Transform

This question really intrigues me, I understand to find the Laplace Transform of Tn, you integrate Tn x exp(-st) from the limits of infinity to zero { ∫∞^0 T n × e^(-st) }Please excuse the bad formatting. But based on this question I am confused as to how to find the function of Tn. Could someone please explain?

Thanks