## apple_pi 3 years ago 1/89 (fibonacci)

1. apple_pi

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

I think by getting a general term....

3. Rowan

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

you also know that the fibonacci series can be generalized by: T[n+2] = T[n] + T[n+1]

5. Lizzardo

So, T[n] = {T[n+2] - T[n+1]} * 10^(n-2)

6. Rowan

Sorry I made a mistake, Lizzardo is right :)

7. mukushla

are u familiar with this formula? $\frac{1}{1-x-x^2}=\sum_{n=0}^{\infty } F_n x^n$

8. apple_pi

er, no

9. phi

when in doubt, try wikipedia see http://en.wikipedia.org/wiki/Fibonacci_number#Power_series

10. apple_pi

@phi this whole thing hinges upon you :D

11. mukushla

@apple_pi now how would u solve this?

12. apple_pi

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

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

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

the formula in wiki starts at F0 =0 F1= 1 F2= 1 F3= 2 and so on

16. apple_pi

Ok thanks