1/89 (fibonacci)

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1/89 (fibonacci)

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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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?
I think by getting a general term....
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

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you also know that the fibonacci series can be generalized by: T[n+2] = T[n] + T[n+1]
So, T[n] = {T[n+2] - T[n+1]} * 10^(n-2)
Sorry I made a mistake, Lizzardo is right :)
are u familiar with this formula? \[\frac{1}{1-x-x^2}=\sum_{n=0}^{\infty } F_n x^n\]
er, no
  • phi
when in doubt, try wikipedia see http://en.wikipedia.org/wiki/Fibonacci_number#Power_series
@phi this whole thing hinges upon you :D
@apple_pi now how would u solve this?
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?
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
  • phi
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
  • phi
the formula in wiki starts at F0 =0 F1= 1 F2= 1 F3= 2 and so on
Ok thanks

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