How to find the sum $\sum_{i = 0}^{\infty}\frac{F_i}{7^i}$?

Question

How to find the sum $\sum_{i = 0}^{\infty}\frac{F_i}{7^i}$?

$F_i$ - $i$-th Fibonacci number

My solution:

I think that that's right to use generating functions.

For Fibonacci number the generating function looks like $\mathcal{F}(z) = \frac{z}{-z^2 - z + 1} $ For the sequence $7^n$ the generating function is $\mathcal{A}(z) = \frac{1}{1 - 7 z}$. But I can't guess now how to use my results to find given sum.

Answer 1

Without generating function:

We establish the recurrence

$$\frac{F_i}{7^i}=\frac{F_{i-1}+F_{i-2}}{7^i}=\frac17\frac{F_{i-1}}{7^{i-1}}+\frac1{49}\frac{F_{i-2}}{7^{i-2}}.$$

Then we extend it to the summation

$$\sum_{i=2}^\infty \frac{F_i}{7^i}=\frac17\sum_{i=2}^\infty \frac{F_{i-1}}{7^{i-1}}+\frac1{49}\sum_{i=2}^\infty \frac{F_{i-2}}{7^{i-2}},$$

giving $$S-\frac{F_1}{7^1}-\frac{F_0}{7^0}=\frac17(S-\frac{F_0}{7^0})+\frac1{49}S,$$ so that

$$S(1-\frac17-\frac1{49})=\frac17,\\S=\frac7{41}.$$

Answer 2

You know $\frac{z}{-z^2 - z + 1} = \mathcal{F}(z) = \sum_{i = 0}^{\infty} F_i z^i$ for any $z$. In particular for $z=1/7$: $$ \sum_{i = 0}^{\infty}\frac{F_i}{7^i} = \mathcal{F}(1/7) = \frac{1/7}{-(1/7)^2 - 1/7 + 1}. $$