Sum of Sequence of Products of Consecutive Fibonacci Numbers

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Theorem

Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers

$\ds \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$


Sum of Even Sequence of Products of Consecutive Fibonacci Numbers

$\ds \sum_{j \mathop = 1}^{2 n} F_j F_{j + 1} = {F_{2 n + 1} }^2 - 1$