I've read this question on Math SE, but I cannot figure out the following:
How to simplify $S$ to be $S=2-\frac{1}{a_{101}}$.
2026-03-26 16:22:58.1774542178
Simplification of a telescopic series
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Since $a_{n+1}=a_n (a_n+1)$, it holds: $\frac{1}{1+a_n}=\frac{a_n}{a_{n+1}}=\frac{1}{a_n}-\frac{1}{a_{n+1}}$. Therefore $S=\sum_{n=1}^{100} \frac{1}{1+a_n}=\sum_{n=1}^{100} \frac{1}{a_n}-\frac{1}{a_{n+1}}=\frac{1}{a_1}-\frac{1}{a_{101}}=2-\frac{1}{a_{101}}$.