HMMT November 2021 U2

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 $100 m+n$.