Deducing an answer on whether n is odd or even

Question

How can I deduce the following:

\begin{align*} F_{n}{^2} - F_{n+1}* F_{n-1}= +1 \text{ or} -1 \end{align*}

Just from knowing if $n$ is even or odd?

I have worked out by hand that if $n$ is odd, the result is +1 and if $n$ is even the result is -1, but I need to know how to work it out without just doing it.

I was given this beforehand and it may or may not be related:

\begin{align*} F_{n+1}{^2} - F_{n+2}* F_{n}= F_{n+1}*F_{n-1}-F_{n}{^2} \end{align*}

From what you were given beforehand you could have constructed a trivial induction proof, if only you tried. — TMM, Apr 17 '17 at 20:32

Jack D'Aurizio · Accepted Answer · 2017-04-17T20:00:00.187

2

Hint:

$$ F_n^2-F_{n+1} F_{n-1} = -\det\begin{pmatrix}F_{n+1} & F_{n} \\ F_{n} & F_{n-1} \end{pmatrix}=-\det\begin{pmatrix}1 & 1 \\ 1 & 0\end{pmatrix}^n = (-1)^{n+1}.$$

Alternative Hint : If we set $\Delta_n = F_n^2-F_{n+1}F_{n-1}$ we have $$ \Delta_{n+1}=F_{n+1}^2-(F_{n+1}+F_n)F_n = F_{n+1}(F_{n+1}-F_n)-F_n^2 = -\Delta_n.$$

edited Apr 17 '17 at 20:00

answered Apr 17 '17 at 19:49

Jack D'Aurizio

353,855

Hmm, we haven't done determinants yet, so not sure. Thanks anyway... – Beastly Gerbil Apr 17 '17 at 19:54
@BeastlyGerbil: no problem, alternative way added. – Jack D'Aurizio Apr 17 '17 at 20:00
How did you arrive to the $\left(\begin{array}{cc}1 & 1 \ 1 & 0\end{array}\right)$ matrix ?. – Felix Marin Apr 17 '17 at 23:24
@FelixMarin: it is more practical to go the other way around, i.e. to prove the powers of such matrix are related with Fibonacci number through induction. – Jack D'Aurizio Apr 18 '17 at 00:16
@JackD'Aurizio I got it. Thanks. – Felix Marin Apr 18 '17 at 00:18

Deducing an answer on whether n is odd or even

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