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OlympiadBench / 2193 / Let m>1 be an integer. A sequence a₁,a₂,a₃,… is defined by…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

answer type

Numerical

is multiple answer

true

language

English

question

Let

m>1

be an integer. A sequence

a_{1}, a_{2}, a_{3}, \ldots

is defined by

a_{1}=a_{2}=1

a_{3}=4

, and for all

n \geq 4

a_{n}=m\left(a_{n-1}+a_{n-2}\right)-a_{n-3} .

Determine all integers

m

such that every term of the sequence is a square.

Plain-text mathematical notation (without MathML)

Let m>1 be an integer. A sequence a₁,a₂,a₃,… is defined by a₁=a₂=1, a₃=4, and for all n≥4,

a_(n)=m(a_(n−1)+a_(n−2))−a_(n−3).

Determine all integers m such that every term of the sequence is a square.

Original LaTeX notation

Let $m>1$ be an integer. A sequence $a_{1}, a_{2}, a_{3}, \ldots$ is defined by $a_{1}=a_{2}=1$, $a_{3}=4$, and for all $n \geq 4$,

$$
a_{n}=m\left(a_{n-1}+a_{n-2}\right)-a_{n-3} .
$$

Determine all integers $m$ such that every term of the sequence is a square.

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 2193
task_id: 8caf44a0-7b11-5735-838c-4bdb34db3df0
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--2193
task_revision_id: 2
config / split: OE_TO_maths_en_COMP / train

Official source

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