Fibonacci Number in terms of Larger Fibonacci Numbers
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Theorem
Let $F_k$ be the $k$th Fibonacci number.
Then:
- $\forall m, n \in \Z_{>0} : \paren {-1}^n F_{m - n} = F_m F_{n - 1} - F_{m - 1} F_n$
Proof
\(\ds F_{m - n}\) | \(=\) | \(\ds F_{m + \paren {-n} }\) | Definition of Integer Subtraction | |||||||||||
\(\ds \) | \(=\) | \(\ds F_{m - 1} F_{-n} + F_m F_{-n + 1}\) | Honsberger's Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{n + 1} F_{m - 1} F_n + \paren {-1}^n F_m F_{n - 1}\) | Fibonacci Number with Negative Index | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {-1}^n F_{m - n}\) | \(=\) | \(\ds \paren {-1} F_{m - 1} F_n + F_m F_{n - 1}\) | multiplying both sides by $\paren {-1}^n$ | ||||||||||
\(\ds \) | \(=\) | \(\ds F_m F_{n - 1} - F_{m - 1} F_n\) | simplifying |
$\blacksquare$