HMMT November, 2021

Problem:

There are $8$ lily pads in a pond numbered $1, 2, \ldots, 8$. A frog starts on lily pad $1$. During the $i$-th second, the frog jumps from lily pad $i$ to $i+1$, falling into the water with probability $\frac{1}{i+1}$. The probability that the frog lands safely on lily pad $8$ without having fallen into the water at any point can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n) = 1$. Find $100m + n$.