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Omni-MATH / Given are real numbers x,y. For any pair of real numbers a₀,a₁, define a sequence by a_(n+2)=xa_(n+1)+ya_(n) for n≥0. Suppose that there exists a fixed nonnegative integer m such t…

Problem

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problem

Given are real numbers x,yx, y. For any pair of real numbers a0,a1a_{0}, a_{1}, define a sequence by an+2=xan+1+yana_{n+2}=x a_{n+1}+y a_{n} for n≥0n \geq 0. Suppose that there exists a fixed nonnegative integer mm such that, for every choice of a0a_{0} and a1a_{1}, the numbers am,am+1,am+3a_{m}, a_{m+1}, a_{m+3}, in this order, form an arithmetic progression. Find all possible values of yy.
Plain-text mathematical notation (without MathML)
Given are real numbers x,y. For any pair of real numbers a₀,a₁, define a sequence by a_(n+2)=xa_(n+1)+ya_(n) for n≥0. Suppose that there exists a fixed nonnegative integer m such that, for every choice of a₀ and a₁, the numbers a_(m),a_(m+1),a_(m+3), in this order, form an arithmetic progression. Find all possible values of y.
Original LaTeX notation
Given are real numbers $x, y$. For any pair of real numbers $a_{0}, a_{1}$, define a sequence by $a_{n+2}=x a_{n+1}+y a_{n}$ for $n \geq 0$. Suppose that there exists a fixed nonnegative integer $m$ such that, for every choice of $a_{0}$ and $a_{1}$, the numbers $a_{m}, a_{m+1}, a_{m+3}$, in this order, form an arithmetic progression. Find all possible values of $y$.

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Source and history

task_id
a37558ec-ac13-581c-a6f7-5b14223d5dac
task_key
test--a37558ec-ac13-581c-a6f7-5b14223d5dac
task_revision_id
1
config / split
/ test

Official source

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