<s>[INST] Prove that $\frac{1}{a_1 + 1} + \frac{1}{a_2 + 1} + \dots + \frac{1}{a_n + 1} < 2$ for all $n \ge 1.$ The sequence $a_n$ is defined by $a_1 = \frac{1}{2}$ and $a_n = a_{n - 1}^2 + a_{n - 1}$ for $n \ge 2.$
Prove that $\frac{1}{a_1 + 1} + \frac{1}{a_2 + 1} + \dots + \frac{1}{a_n + 1} < 2$ for all $n \ge 1.$
 [/INST] You have
$$\frac{a_k}{a_{k-1}}=a_{k-1}+1$$ and
$$a_k - a_{k-1}=a_{k-1}^2$$
Hence,
$$\sum_{k=1}^n\frac 1{a_k+1}= \sum_{k=1}^n\frac{a_{k}}{a_{k+1}}= \sum_{k=1}^n\frac{a_{k}^2}{a_{k}a_{k+1}}$$
$$=  \sum_{k=1}^n\frac{a_{k+1}-a_k}{a_{k}a_{k+1}}=\sum_{k=1}^n\left(\frac{1}{a_{k}} - \frac 1{a_{k+1}}\right)$$
I leave the last step up to you.
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