Sum of Sequence of Odd Index Fibonacci Numbers

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

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

That is:
 * $F_1 + F_3 + F_5 + \cdots + F_{2n-1} = F_{2n}$

Proof
Proof by induction:

For all $n \in \N_{>0}$, let $P \left({n}\right)$ be the proposition:
 * $\displaystyle \sum_{j \mathop = 1}^n F_{2j - 1} = F_{2n}$

Basis for the Induction
$P(1)$ is the case $F_1 = 1 = F_2$, which holds from the definition of Fibonacci numbers.

This is our basis for the induction.

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

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

Then we need to show:
 * $\displaystyle \sum_{j \mathop = 1}^{k+1} F_{2j - 1} = F_{2k + 2}$

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:
 * $\displaystyle \forall n \ge 1: \sum_{j \mathop = 1}^n F_{2j - 1} = F_{2n}$.