A Reciprocal-Arithmetic Family

★ 7/10Arithmetic Progression sequences Estimation Telescoping

Problem

A positive sequence $\{a_n\}$ satisfies $a_1 = 1$ and

a_{n+2}\left(a_{n+1} - a_n\right) = a_n\left(a_{n+2} - a_{n+1}\right), \qquad n \in \mathbb{N}^*.

Let $T_n = a_1a_2 + a_2a_3 + \cdots + a_na_{n+1}$ , and suppose $T_{12} = 4$ . Determine, with proof, which of the following are true:

1. $\{a_n\}$ is decreasing; 2. $a_{2024} = \dfrac{6}{2029}$ ; 3. there exists $n$ with $T_n = \dfrac{4}{3}$ ; 4. $\displaystyle\sum_{i=1}^{100} a_i > 10$ .

Answer

Solution

Difficulty	7/10
Topics	Arithmetic Progression, sequences, Estimation, Telescoping

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