Hidden Telescoping Sum – Mattia Giuri's bizarre blog

Here’s an exercise I proposed in an online competition.

You have the following sequence:

$a_1 = \frac{1}{70}, a_{n+1} = a_n^2 + a_n$

Find the integer part of:

$\sum_{i=1}^{100} \frac{1}{a_i + 1}$

Now, notice that

$\frac{1}{a_{n+1}} = \frac{1}{a_n(a_n + 1)} = \frac{1}{a_n} - \frac{1}{a_n+1}$

Hence

$\frac{1}{a_n} - \frac{1}{a_{n+1}} = \frac{1}{a_n + 1}$

By substitution, we have that the sum telescopes and reduces to

$\frac{1}{a_1} - \frac{1}{a_{101}}$

Since $a_{101}$ is a very large number (the sequence has a quadratic increasing rate), we have that the integer part of the sum is

$69$