Prove by induction: $$ f_{n-1}^2 - f_{n-2}^2 = f_n f_{n-3} $$ where $f_n$ refers to the $n$'th Fibonacci term.
2026-03-28 09:55:41.1774691741
Prove by mathematical induction the following Fibonacci statement
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Induction is for suckers. ^_^
$$F_n=F_{n-1}+F_{n-2}$$ by the definition of the Fibonacci sequence. Also, $F_{n-1}=F_{n-2}+F_{n-3}$, so $$F_{n-3}=F_{n-1}-F_{n-2}$$ Multiplying those together, we get $$F_nF_{n-3}=(F_{n-1}+F_{n-2})(F_{n-1}-F_{n-2})=F^2_{n-1}-F^2_{n-2}$$