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2018 Putnam B Problems/Problem 4

Revision as of 17:49, 17 August 2025 by Pinotation (talk | contribs) (Created page with "==Problem== Given a real number <math>a</math>, we define a sequence by <math>x_0 = 1</math>, <math>x_1 = x_2 = a</math>, and <math>x_{n+1} = 2x_nx_{n-1} - x_{n-2}</math> for...")
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Problem

Given a real number $a$, we define a sequence by $x_0 = 1$, $x_1 = x_2 = a$, and $x_{n+1} = 2x_nx_{n-1} - x_{n-2}$ for $n \ge 2$. Prove that if $x_n = 0$ for some $n$, then the sequence is periodic.

Solution

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