## Fibonacci Series

#### Problem

Let $S = 1/2 + 1/4 + 2/8 + 3/16 + 5/32 + ... + F_k / 2^k + ...$, where F_{$k$} represents the $k$th term of the Fibonacci sequence: $1, 1, 2, 3, 5, 8, 13, ...$ .

Find the value of $S$.

#### Solution

$\begin {align}S &= F_1 / 2 + F_2 / 4 + F_3 / 8 + F_4 / 16 + ...\\2S &= F_1 + F_2 / 2 + F_3 / 4 + F_4 / 8 + ...\\4S &= 2F1 + F_2 + F_3 / 2 + F_4 / 4 + ...\\\therefore 4S - 2S - S &= 2F_1 + F_2 - F_1 +(F_3 - F_2 - F_1) / 2 + (F_4 - F_3 - F_2) / 4 + ...\end {align}$

But as $F_k = F_{k-1} + F_{k-2}$ it follows that $F_k - F_{k-1} + F_{k-2} = 0$.

$\therefore S = 2F_1 + F_2 - F1 = F_1 + F_2 = 2.

Find the value of $x$ for which $a(x) = x F_1 + x^2 F_2 + x^3 F_3 + ... = 1$.

Prove that $a(x) = G$, where $G$ is a positive integer, always has a solution.

Problem ID: 299 (17 Dec 2006) Difficulty: 3 Star