A game consists of jugling a ball by touching the ball on the screen. The score is the number of jugles. So if $p$ is the probability of touching the ball at the right moment so that it bouces back, what is the probability of beating your hight score $:=n$ i.e. maknig at least $n$ succssful jugles in a row?
I find the value $$\sum\limits_{k=n}^\infty\prod\limits_{m=1}^k p^m$$ because it's the union of "making m successful jugles in a row" with $m\geq n$ assuming those events are independent. Is that true?
Probability of getting a score $k\equiv f(k)$ $$f(k)=p^k(1-p)$$
Now you break the highscore $n$ if you do atleast $n+1$ juggles. $$P(k>n)=\sum_{k=n+1}^\infty p^k(1-p)=\frac{p^{k+1}(1-p)}{1-p}=p^{k+1}$$