<s>[INST] Prove $u^2_n-u_{n-1}u_{n+1}=(-1)^n$ if $u_1=1,u_2=2,u_n=u_{n-1}+u_{n-2},n>2$ Prove $u^2_n-u_{n-1}u_{n+1}=(-1)^n$ if $u_1=1,u_2=2,u_n=u_{n-1}+u_{n-2},n>2$
If $n=3$ equality is true.
How to prove this by induction?
 [/INST] We may use the following famous trick: Notice that
$$ \begin{pmatrix}u_{n+2} \\ u_{n+1}\end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} u_{n+1} \\ u_n \end{pmatrix}. $$
Recursively using this relation, we have
$$ \begin{pmatrix}u_{n+1} & u_n \\ u_{n} & u_{n-1} \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^{n-1} \begin{pmatrix} u_2 & u_1 \\ u_1 & u_0 \end{pmatrix}. $$
Here, we extrapolate the sequence $(u_n)$ using the recurrence relation so that $u_0 = 1$. Now taking the determinant gives the desied result.
</s>