Sum of Even Sequence of Products of Consecutive Fibonacci Numbers

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

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

Proof
From Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers:
 * $(1): \quad \displaystyle \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$

Hence: