Sum of Sequence of Fibonacci Numbers

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

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

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

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:
 * $$\sum_{j=1}^k F_j = F_{k+2} - 1$$

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

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