Sum of Sequence of Fibonacci Numbers

Theorem
Let $F_k$ be the $k$th Fibonacci number.

Then:
 * $\displaystyle \forall n \ge 2: \sum_{j=1}^n F_j = F_{n+2} - 1$

Proof
From the initial definition of Fibonacci numbers, we have:
 * $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$

Proof by induction:

For all $n \in \N^*$, let $P \left({n}\right)$ be the proposition:
 * \displaystyle $\sum_{j=1}^n F_j = F_{n+2} - 1$

Basis for the Induction

 * $P(2)$ is the case $F_1 + F_2 = 2 = F_4 - 1$, which holds.

This is our basis for the induction.

Induction Hypothesis

 * Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:
 * $\displaystyle \sum_{j=1}^k F_j = F_{k+2} - 1$

Then we need to show:
 * $\displaystyle \sum_{j=1}^{k+1} F_j = F_{k+3} - 1$

Induction Step
This is our induction step:

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \forall n \ge 2: \sum_{j=1}^n F_j = F_{n+2} - 1$