I need to show that for the terms of the fibonnaci sequence, where $n \ge 1$, $a_{n-1} \times a_{n+1} = a_n^2 + (-1)^n$
And the formula to calculate fibonacci numbers where $n \ge 1$ is $a_n = a_{n-1} + a_{n-2}$
I've tried subbing in just about everything starting with the left side, then I tried the right side. I can't get it to work through induction. Which side should I start with? And what should I be substituting to make it equal to the otherside?
After covering any base cases, you can do the following:
\begin{eqnarray} a_{n-1} \color{blue}{a_{n+1}} &=& a_{n-1}\color{blue}{(a_n+a_{n-1})} \tag{by definition of $a_{n+1}$}\\ &=& a_{n} a_{n-1} + \color{red}{a_{n-1}^2}\tag{distributing $a_{n-1}$}\\ &=& a_{n} a_{n-1} + \color{red}{a_{n-2} a_n - (-1)^{n-1}} \tag{induction hypothesis}\\ &=& a_n\color{blue}{(a_{n-1} + a_{n-2})} - (-1)^{n-1} \tag{factoring out $a_n$}\\ &=& a_n \color{blue}{a_n} - (-1)^{n-1} \tag{by definition of $a_n$}\\ &=& a_n^2 + (-1)^n \end{eqnarray}