n-0-1-n-F-n-7-n-F-n-denotes-Fibbonocci-sequence-

Question Number 122023 by Dwaipayan Shikari last updated on 13/Nov/20

\underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n} F_{n}}{7^{n}} (F_{n} d e n o t e s F i b b o n o c c i s e q u e n c e)

Answered by mr W last updated on 13/Nov/20

F_{n} = \frac{1}{\sqrt{5}} (ϕ^{n} - ψ^{n}) w i t h

ϕ = \frac{1 + \sqrt{5}}{2}

ψ = \frac{1 - \sqrt{5}}{2}

S = \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n} F_{n}}{7^{n}}

= \frac{1}{\sqrt{5}} \underset{n = 0}{\sum^{\infty}} [{(- \frac{ϕ}{7})}^{n} - {(- \frac{ψ}{7})}^{n}]

= \frac{1}{\sqrt{5}} (\frac{1}{1 + \frac{ϕ}{7}} - \frac{1}{1 + \frac{ψ}{7}})

= \frac{1}{\sqrt{5}} (\frac{1}{1 + \frac{1 + \sqrt{5}}{14}} - \frac{1}{1 - \frac{\sqrt{5} - 1}{14}})

= \frac{14}{\sqrt{5}} (\frac{1}{15 + \sqrt{5}} - \frac{1}{15 - \sqrt{5}})

= - \frac{7}{55}

Commented by Dwaipayan Shikari last updated on 13/Nov/20

T y p o \frac{1}{1 - \frac{\sqrt{5} - 1}{14}} = \frac{14}{15 - \sqrt{5}}

Commented by Dwaipayan Shikari last updated on 14/Nov/20

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