A Markov-Style Sequence

★ 7/10number theory Recursion sequences Vieta's Formulas

Problem

A sequence $\{a_n\}$ satisfies $a_0 = 0$ and

a_n = \frac{1}{2}\left(3a_{n-1} + \sqrt{5a_{n-1}^2 + 4}\right), \quad n \geqslant 1.

Prove that every term of the sequence is a natural number, and that for any three consecutive terms $a_n, a_{n+1}, a_{n+2}$ , each pairwise product plus $1$ is a perfect square.

Answer

Solution

Difficulty	7/10
Topics	number theory, Recursion, sequences, Vieta's Formulas

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments

No comments yet.