Prove by Mathematical Induction that: $$a^2_{n+1} - a_{n}a_{n+2} = (-1)^n$$
Here the terms are from the Fibonacci Sequence.
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Shivam Vishwekar
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1Welcome to MSE. Your math expressions seems screwed up. – macton Jan 29 '21 at 07:43
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1https://math.stackexchange.com/questions/523925/induction-proof-on-fibonacci-sequence-fn-1-cdot-fn1-fn2-1n, https://math.stackexchange.com/questions/20948/fibonacci-identity-f-n-1f-n1-f-n2-1n?noredirect=1&lq=1, https://math.stackexchange.com/questions/1420281/fibonacci-number-identity?noredirect=1&lq=1,https://math.stackexchange.com/questions/2415088/proof-cassinis-identity-with-induction-and-fibonacci-sequence, https://proofwiki.org/wiki/Cassini%27s_Identity. – player3236 Jan 29 '21 at 07:47
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Are you stuck on the base step or the inductive step? If you edit in what you managedso far, we can discuss the next part. – J.G. Jan 29 '21 at 08:16
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I was stucked in the inductive step. – Shivam Vishwekar Jan 29 '21 at 11:41
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A hint: let $a:=F_n,\,b:=F_{n+1}$ so$$\color{blue}{F_{n+2}^2-F_{n+1}F_{n+3}}+\color{red}{F_{n+1}^2-F_nF_{n+2}}=(a+b)^2-b(a+2b)+b^2-a(a+b).$$Simplify this.
J.G.
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@ShivamVishwekar I've added some colour. The inductive hypothesis assumes a value for the red terms, from which you can deduce a value for the blue terms. – J.G. Jan 29 '21 at 12:13
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