## The Fibonacci Series

#### Problem

Given that $F_{n}$ represents the $n^{th}$ term of the Fibonacci sequence: $1, 1, 2, 3, 5, 8, 13, ...,$ and $S_{n} = F_{1} + F_{2} + ... + F_{n},$ prove that $S_{n} = F_{n+2} - 1$.

For example, $S_5 = F_1 + F_2 + ... + F_5 = 1 + 1 + 2 + 3 + 5 = 12 = 13 - 1 = F_7 - 1$.

Problem ID: 352 (17 Apr 2009) Difficulty: 3 Star