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Omni-MATH / A sequence a₁,a₂,a₃,… of positive reals satisfies a_(n+1)=√((1+a_(n))/(2)). Determine all a₁ such that a_(i)=(√(6)+√(2))/(4) for some positive integer i.

Problem

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problem

A sequence a1,a2,a3,…a_{1}, a_{2}, a_{3}, \ldots of positive reals satisfies an+1=1+an2a_{n+1}=\sqrt{\frac{1+a_{n}}{2}}. Determine all a1a_{1} such that ai=6+24a_{i}=\frac{\sqrt{6}+\sqrt{2}}{4} for some positive integer ii.
Plain-text mathematical notation (without MathML)
A sequence a₁,a₂,a₃,… of positive reals satisfies a_(n+1)=√((1+a_(n))/(2)). Determine all a₁ such that a_(i)=(√(6)+√(2))/(4) for some positive integer i.
Original LaTeX notation
A sequence $a_{1}, a_{2}, a_{3}, \ldots$ of positive reals satisfies $a_{n+1}=\sqrt{\frac{1+a_{n}}{2}}$. Determine all $a_{1}$ such that $a_{i}=\frac{\sqrt{6}+\sqrt{2}}{4}$ for some positive integer $i$.

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Source and history

task_id
ceaefea8-ade3-5820-a9f4-dd48e492281f
task_key
test--ceaefea8-ade3-5820-a9f4-dd48e492281f
task_revision_id
1
config / split
/ test

Official source

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