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apple_pi

  • 3 years ago

1/89 (fibonacci)

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  1. apple_pi
    • 3 years ago
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    If you sum all the fibonacci numbers like this: 1 * 10^-2 + 1 * 10^-3 + 2 * 10^-4 + 3 * 10^-5 + 5 * 10^-6 + 8 * 10^-7 + ... You end up getting 1/89. How can this be proven?

  2. Lizzardo
    • 3 years ago
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    I think by getting a general term....

  3. Rowan
    • 3 years ago
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    This is called a geometric series (which are fortunately convergent). There is a formula s = 1/(1-r) where r is the ratio of the n+1 th term divided by the nth term

  4. Lizzardo
    • 3 years ago
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    you also know that the fibonacci series can be generalized by: T[n+2] = T[n] + T[n+1]

  5. Lizzardo
    • 3 years ago
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    So, T[n] = {T[n+2] - T[n+1]} * 10^(n-2)

  6. Rowan
    • 3 years ago
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    Sorry I made a mistake, Lizzardo is right :)

  7. mukushla
    • 3 years ago
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    are u familiar with this formula? \[\frac{1}{1-x-x^2}=\sum_{n=0}^{\infty } F_n x^n\]

  8. apple_pi
    • 3 years ago
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    er, no

  9. phi
    • 3 years ago
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    when in doubt, try wikipedia see http://en.wikipedia.org/wiki/Fibonacci_number#Power_series

  10. apple_pi
    • 3 years ago
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    @phi this whole thing hinges upon you :D

  11. mukushla
    • 3 years ago
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    @apple_pi now how would u solve this?

  12. apple_pi
    • 3 years ago
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    in this case x = 0.1 so sum = 1/ (1-0.1-0.01) = 1/0.89 = 100/89 = 1.1235955... So do we divide by 100? and where did that come from?

  13. mukushla
    • 3 years ago
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    note that what u got is 1 * + 1 * 10^-1 + 2 * 10^-2 + 3 * 10^-3 + 5 * 10^-4 + 8 * 10^-5 + ... multiply it by 10^-2 to get ur answer

  14. phi
    • 3 years ago
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    First, the article derives the formula muk posted. but there is supposed to be an x up top which he left out. also, for your sequence, first factor a 0.1 out of your numbers, so that it matches the formula

  15. phi
    • 3 years ago
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    the formula in wiki starts at F0 =0 F1= 1 F2= 1 F3= 2 and so on

  16. apple_pi
    • 3 years ago
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    Ok thanks

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