Assuming F represents the Fibonacci sequence, use mathematical induction to prove that for all integers n >= 0, Fsub(n+2)Fsub(n) − (Fsub(n+1))^2 = (−1)^n.

I have gotten to the inductive goal when plugging in k+1, but am lost from here.

Thanks.

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- Nov 14th 2012, 01:10 PMWalshyFibonacci Proof by Induction
Assuming F represents the Fibonacci sequence, use mathematical induction to prove that for all integers n >= 0, Fsub(n+2)Fsub(n) − (Fsub(n+1))^2 = (−1)^n.

I have gotten to the inductive goal when plugging in k+1, but am lost from here.

Thanks. - Nov 14th 2012, 01:33 PMMacstersUndeadRe: Fibonacci Proof by Induction
Assuming, (after confirming base case)

F_{k+2}F_{k}− (F_{(k+1)}^{2}= (−1)^{k}. (Assumption P)

With the induction step, you want to prove

F_{k+3}F_{k+1}- (F_{k+2})^{2}= (-1)^{k+1}

so,

(-1)^{k+1}= (-1)^{k}*(-1)^1 =**-1 * ( F**) (by Assumption P)_{k+2}F_{k}− (F_{(k+1)}^{2}

Can you take it from here? I currently don't have pencil and paper handy to give you the full answer.

You'll probably also have to use the fact that F_{k+2}= F_{k}+ F_{k+1}

For simplicity, I would leave the bolded and then work in the other direction as well

In other words, working with F_{k+3}F_{k+1}- (F_{k+2})^{2}so that it's equal to what is in the bold, since A = B = C implies A = C - Nov 14th 2012, 02:05 PMWalshyRe: Fibonacci Proof by Induction
Yeah that makes sense, thanks. I'm in the process of trying to figure out the rest, will update.

- Nov 14th 2012, 02:19 PMMacstersUndeadRe: Fibonacci Proof by Induction
I look forward to it. This problem did look very familiar from a number theory course I took.

- Nov 14th 2012, 03:33 PMWalshyRe: Fibonacci Proof by Induction
Yeah i figured it out. Thanks for all the help.

- Nov 14th 2012, 03:36 PMMacstersUndeadRe: Fibonacci Proof by Induction
No problem. By curiosity was there anything I missed noting that helped you solve the problem?

- Nov 14th 2012, 04:13 PMWalshyRe: Fibonacci Proof by Induction
No i don't think so. Your explanation was very thorough, and as u said, u didnt have a pencil and paper to work out the last part, but that wasn't too bad.