Math Induction Proof

Question

I'm having a little bit of trouble with this proof.

Use mathematical induction to show that $f_{n−1} \cdot f_{n+1} − f_{2 n} = (−1)^n$ for $n$ in the set of positive integers.

I know that in recursive functions, at least the first term is provided and then you could play around with it to come to a conclusion.

I was maybe thinking of using the Fibonacci numbers definition as a guide but I'm not really going forward.

$f_n$ is the FIbonacci sequence. Does the identity involve $f_{2n}$ or $f_n^2$? — grand_chat, Nov 30 '16 at 01:05

score 0 · Answer 1 · answered Nov 30 '16 at 01:29

0

Hint $\ $ Take the determinant of $$ \left(\begin{array}{ccc} 1 & 1 \\\ 1 & 0 \end{array}\right)^n\ =\ \left(\begin{array}{ccc} f_{n+1} & f_n \\\ f_n & f_{n-1} \end{array}\right) $$

answered Nov 30 '16 at 01:29

Bill Dubuque

272,048

If @jake0991 is just starting off with proof by induction it is unlikely that they know enough about determinants to make this useful. – Q the Platypus Nov 30 '16 at 03:02
@QthePlatypus It's actually quite simple. If anyone has questions then ask and I will elaborate. – Bill Dubuque Nov 30 '16 at 03:08

Math Induction Proof

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