let-be-the-sequence-x-n-n-1-defined-by-x-1-1-x-n-2-3x-n-1-x-n-n-N-find-L-lim-n-k-0-0-x-2k-1-x-k-x-k-1-1-n-

Question Number 219561 by hardmath last updated on 28/Apr/25

$$\mathrm{let}\:\mathrm{be}\:\mathrm{the}\:\mathrm{sequence}\:\:\:\left(\mathrm{x}_{\boldsymbol{\mathrm{n}}} \right)\mathrm{n}\:\geqslant\:\mathrm{1} \\ $$$$\mathrm{defined}\:\mathrm{by}\:\:\:\mathrm{x}_{\mathrm{1}} =\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}_{\boldsymbol{\mathrm{n}}+\mathrm{2}} \:=\:\mathrm{3x}_{\boldsymbol{\mathrm{n}}+\mathrm{1}} −\:\mathrm{x}_{\boldsymbol{\mathrm{n}}} \\ $$$$\forall\mathrm{n}\:\in\:\mathbb{N} \\ $$$$\mathrm{find}\:\:\:\boldsymbol{\mathrm{L}}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\sqrt[{\boldsymbol{\mathrm{n}}}]{\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\mathrm{0}} {\sum}}\:\:\frac{\mathrm{x}_{\mathrm{2}\boldsymbol{\mathrm{k}}+\mathrm{1}} }{\mathrm{x}_{\boldsymbol{\mathrm{k}}} \:+\:\mathrm{x}_{\boldsymbol{\mathrm{k}}+\mathrm{1}} }}\:=\:? \\ $$

Commented by Nicholas666 last updated on 28/Apr/25

$$\frac{\mathrm{7}+\mathrm{3}\sqrt{\mathrm{5}}}{\mathrm{2}}\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Commented by mr W last updated on 30/Apr/25

$${the}\:{sequence}\:{is}\:{not}\:{sufficiently} \\ $$$${defined}\:{with}\:{x}_{\mathrm{1}} =\mathrm{1}\:{alone}.\: \\ $$$${for}\:{example}\:{you}\:{can}\:{not}\:{determine}\: \\ $$$${x}_{\mathrm{2}} =?. \\ $$$${please}\:{recheck}\:{the}\:{question}! \\ $$

Commented by hardmath last updated on 02/May/25

$$\mathrm{thankyou}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{prifessor} \\ $$

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