Ideas

• Mathematical induction

Claim

Let $(f_n)$ be the sequence of Fibonacci numbers, i.e.

Let $(s_n) = \sum_{n}^{\infty} f_n$. Then the following statement is true:

Proof

For $n=1$, $s_1 = f_1 = 1$. And we can know it satisfies the claim, since $f_3 - 1 = 2 - 1 = 1$.

Assume that the claim is true for some $n=k, k \in \mathbb{N}$:

We need to show that it also holds for $n=k+1$:

From the claim:

Adding $f_{k+1}$ to both sides:

Using the definition of Fibonacci numbers, this can be written as:

This completes the inductive step. By the principle of mathematical induction, the claim is true for all $n \in \mathbb{N}$.