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


Solution

From the definition, $F_{n} = F_{n-1} + F_{n-2},$ we get $F_{n-2} = F_{n} - F_{n-1}$.

$$\begin{aligned}\therefore S_{n} &= F_{1} + F_{2} + ... + F_{n}\\&= (F_{3} - F_{2}) + (F_{4} - F_{3}) + ... + (F_{n+1} - F_{n}) + (F_{n+2} - F_{n+1})\\&= F_{n+2} - F_{2}\\&= F_{n+2} - 1\end{aligned}$$
Problem ID: 352 (17 Apr 2009)     Difficulty: 3 Star

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