Fibonacci Number in terms of Larger Fibonacci Numbers

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

$\blacksquare$