Question

1/89 (fibonacci)

Answer 1

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?

Answer 2

I think by getting a general term....

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

Sorry I made a mistake, Lizzardo is right :)

Answer 7

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

Answer 8

er, no

Answer 9

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

Answer 10

@phi this whole thing hinges upon you :D

Answer 11

@apple_pi

now how would u solve this?

Answer 12

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?

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

Ok thanks