How Does 1/89 Relate to Fibonacci Numbers in Its Decimal Expansion?

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The decimal expansion of 1/89 is 0.01123595..., which directly relates to Fibonacci numbers through the equation 1/10 * Σ(F_n / 10^n) = 1/89, where F_n represents the nth Fibonacci number. The generating function for Fibonacci numbers, g(x) = Σ(f_n * x^n) = x / (1 - x - x^2), can be evaluated at x = 1/10 to yield Σ(f_n * 10^(-n)) = 10/89. This relationship highlights the deep connection between Fibonacci sequences and rational numbers.

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\text{We have: }\:\dfrac{1}{89} \;=\;0.01123595\,\,.\,.\,.


\text{The decimal is formed like this:}

. . 0.0{\bf1}
. . 0.00{\bf1}
. . 0.000{\bf2}
. . 0.0000{\bf3}
. . 0.00000{\bf5}
. . 0.000000{\bf8}
. . 0.000000{\bf{13}}
. . 0.0000000{\bf{21}}
. . 0.00000000{\bf{34}}
. . . . . . \vdots


\displaystyle\text{It seems that: }\:\frac{1}{10}\sum^{\infty}_{n=1} \frac{F_n}{10^n} \;=\;\frac{1}{89}

. . \text{where }F_n\text{ is the }n^{th}\text{ Fibonacci number.}


\text{Care to prove it?}
 
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soroban said:
\text{We have: }\:\dfrac{1}{89} \;=\;0.01123595\,\,.\,.\,.


\text{The decimal is formed like this:}. . 0.0{\bf1}
. . 0.00{\bf1}
. . 0.000{\bf2}
. . 0.0000{\bf3}
. . 0.00000{\bf5}
. . 0.000000{\bf8}
. . 0.000000{\bf{13}}
. . 0.0000000{\bf{21}}
. . 0.00000000{\bf{34}}
. . . . . . \vdots


\displaystyle\text{It seems that: }\:\frac{1}{10}\sum^{\infty}_{n=1} \frac{F_n}{10^n} \;=\;\frac{1}{89}

. . \text{where }F_n\text{ is the }n^{th}\text{ Fibonacci number.}


\text{Care to prove it?}

The Fibonacci's numbers have been studied for something like 800 years and, among the others 'discoveries' there is the the generating function that can be directly derived from the difference equation $\displaystyle f_{n+2}= f_{n+1}+f_{n},\ f_{0}=0,\ f_{1}=1$...

$\displaystyle g(x)=\sum_{n=1}^{\infty} f_{n}\ x^{n} = \frac{x}{1-x-x^{2}}$ (1)

Setting in (1) $x=\frac{1}{10}$ You have...

$\displaystyle \sum_{n=1}^{\infty} f_{n}\ 10^{- n} = \frac{10}{89}$ (2)

Kind regards

$\chi$ $\sigma$
 
chisigma said:
The Fibonacci's numbers have been studied for something like 800 years and, among the others 'discoveries' there is the the generating function that can be directly derived from the difference equation $\displaystyle f_{n+2}= f_{n+1}+f_{n},\ f_{0}=0,\ f_{1}=1$...

$\displaystyle g(x)=\sum_{n=1}^{\infty} f_{n}\ x^{n} = \frac{x}{1-x-x^{2}}$ (1)

Setting in (1) $x=\frac{1}{10}$ You have...

$\displaystyle \sum_{n=1}^{\infty} f_{n}\ 10^{- n} = \frac{10}{89}$ (2)

... but much more 'elegant' is what You obtain setting in (1) $x=\frac{1}{2}$...

$\displaystyle \sum_{n=1}^{\infty} \frac{f_{n}}{2^{n}} = 2$

Not bad!(Happy)...

Kind regards

$\chi$ $\sigma$