A Telescoping Reciprocal Sum

★ 6/10Recursion sequences Telescoping inequality

Problem

A sequence $\{a_n\}$ satisfies $a_1 = \dfrac{4}{3}$ and

a_{n+1} - 1 = a_n^2 - a_n, \quad n \in \mathbb{N}^*.

Let $S_n$ be the sum of the first $n$ terms of $\left\{\dfrac{1}{a_n}\right\}$ . Determine the pair of consecutive integers between which $S_{2021}$ lies, with proof.

Answer

Solution

Difficulty	6/10
Topics	Recursion, sequences, Telescoping, inequality

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