Sum of Even Sequence of Products of Consecutive Fibonacci Numbers
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Theorem
Let $F_k$ be the $k$'th Fibonacci number.
Then:
- $\ds \sum_{j \mathop = 1}^{2 n} F_j F_{j + 1} = {F_{2 n + 1} }^2 - 1$
Proof
From Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers:
- $(1): \quad \ds \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$
Hence:
\(\ds \sum_{j \mathop = 1}^{2 n} F_j F_{j + 1}\) | \(=\) | \(\ds \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} + F_{2 n} F_{2 n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {F_{2 n} }^2 + F_{2 n} F_{2 n + 1}\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds F_{2 n} \paren {F_{2 n} + F_{2 n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds F_{2 n} F_{2 n + 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {F_{2 n + 1} }^2 - 1\) | Cassini's Identity |
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {1-1}$ Principle of Mathematical Induction: Exercise $12$