A Radical Recurrence

★ 7/10Substitution sequences Telescoping

Problem

A sequence has $a_1 = 2$ and

a_{n+1} = \frac19\left(a_n + 2\sqrt{4a_n + 1} + 2\right).

1. Find a formula for $a_n$ . 2. Let $T_n$ be the sum of the first $n$ terms of $\left\{\dfrac{1}{a_n}\right\}$ . Prove that

T_n > \frac{4n}{3} - \frac{14}{9}.

Answer

Solution

Difficulty	7/10
Topics	Substitution, sequences, Telescoping

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