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Given the sequence \(\frac{1}{3}+\frac{1}{9}+\frac{2}{27}+\frac{3}{81}+.....\frac{F_k}{3^k}+......\) where \(F_k\) is the Fibonacci sequence. Compute the sum.

Note by William Isoroku 2 years, 7 months ago

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Here, Let S be its sum.

So, \( {S} - \frac{S}{3} = \frac{1}{3} - \frac{1}{3^2} {(S)} \)

Or \( S= \frac{3}{5} \)

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Use this \[\sum_{k=1}^{\infty} F_k x^k = \frac{x}{1-x-x^2} \quad \forall |x| < \frac{\sqrt5-1}{2}\]

@William Isoroku

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Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

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TopNewestHere, Let

Sbe its sum.So, \( {S} - \frac{S}{3} = \frac{1}{3} - \frac{1}{3^2} {(S)} \)

Or \( S= \frac{3}{5} \)

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Use this \[\sum_{k=1}^{\infty} F_k x^k = \frac{x}{1-x-x^2} \quad \forall |x| < \frac{\sqrt5-1}{2}\]

@William Isoroku

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