A Fixed-Point Transformation

★ 7/10Recursion sequences Induction Telescoping

Problem

In a sequence $\{a_n\}$ , $a_1 = 2$ and

a_{n+1} = \left(\sqrt{2} - 1\right)(a_n + 2), \qquad n = 1, 2, 3, \ldots

1. Find a formula for $a_n$ . 2. A sequence $\{b_n\}$ has $b_1 = 2$ and

b_{n+1} = \frac{3b_n + 4}{2b_n + 3}.

Prove that $\sqrt{2} < b_n \leqslant a_{4n-3}$ for all $n$ .

Answer

Solution

Difficulty	7/10
Topics	Recursion, sequences, Induction, Telescoping

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