Question 55
Suppose $a_{1}, a_{2}, \ldots, a_{100}$ are positive real numbers such that
$$
a_{k}=\frac{k a_{k-1}}{a_{k-1}-(k-1)}
$$
for $k=2,3, \ldots, 100$. Given that $a_{20}=a_{23}$, compute $a_{100}$.
$$
a_{k}=\frac{k a_{k-1}}{a_{k-1}-(k-1)}
$$
for $k=2,3, \ldots, 100$. Given that $a_{20}=a_{23}$, compute $a_{100}$.