Fibonacci Decimal

Level 2

∑ n = 1 ∞ F n 10 0 n = 0.01010203050813213455... = a b \sum_{n=1}^{\infty} \frac{F_n}{100^n} = 0.01010203050813213455... = \frac{a}{b}

where F n is the Fibonacci sequence and gcd ⁡ ( a , b ) = 1 . Find b . \text{where } F_n \text{ is the Fibonacci sequence and } \gcd(a,b) = 1 \text{ . Find } b .


The answer is 9899.

Albert Yiyi by albert yiyi , 31, Malaysia

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1 solution

Albert Yiyi
Jun 8, 2018

100 S = 1. 01 02 03 05 08 13 21... S = 0. 01 01 02 03 05 08 13... 99 S = 1. 00 01 01 02 03 05 08... 9900 S = 100. 01 01 02 03 05 08 13... S = 0. 01 01 02 03 05 08 13... 9899 S = 100. 00 00 00 00 00 00 00 S = 100 9899 ∴ b = 9899 \begin{aligned} 100S &= 1.\ 01\ 02\ 03\ 05\ 08\ 13\ 21... \\ S &= 0.\ 01\ 01\ 02\ 03\ 05\ 08\ 13... \\ 99S &= 1.\ 00\ 01\ 01\ 02\ 03\ 05\ 08... \\ \\ 9900S &= 100.\ 01\ 01\ 02\ 03\ 05\ 08\ 13... \\ S &= \> \> \> \> 0.\ 01\ 01\ 02\ 03\ 05\ 08\ 13... \\ 9899S &= 100.\ 00\ 00\ 00\ 00\ 00\ 00\ 00 \\ \\ S &= \frac{100}{9899} \\ \therefore b &= 9899 \end{aligned}

5 Helpful 2 Interesting 0 Brilliant 0 Confused

This looks fishy. Are you saying that all Fibonacci numbers are 2-digit integers, so there's no carry over from the expression 100 S − S 100S - S ?

Pi Han Goh - 3 years ago

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2-digit, carry over, those are not relevant.

the solution presented is still valid, but not rigorous.

there's a rigorous way, here's a hint if u wanna try: let (1/100) = x

S = 1x + 1x^2 + 2x^3 + 3x^4+ 5x^5 + 8x^6 + ...

albert yiyi - 3 years ago

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Yup, that's correct !

Pi Han Goh - 3 years ago

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