I have a sequence defined as:
$u_1 = 1$
$u_2 = 1$
$u_{k+1} = u_{k-1} + u_k$
Which gives: $$1, 1, 2, 3, 5, 8\ldots$$
I need to prove for $n \geqslant 2$ that:
$$u_n^2 = u_{n-1}\cdot u_{n+1} + (-1)^{n-1}$$
So for the base case:
$$1^2 = 1*2 + (-1)^1$$ $$1^2 = 1$$
Then for the induction hypothesis I have:
$$u^2_{k} = u_{k-1}\cdot u_{k+1} + (-1)^{k-1}$$
So I need to show that $${u^2_{k+1}} = u_{k}\cdot u_{k+2} + (-1)^{k}$$
I started by multiplying the induction hypothesis by $-1$
$$u_{k-1}\cdot u_{k+1} - u^2_{k} = (-1)^k$$
but I got stuck at this point, have I structured this in the wrong way?
EDIT:Others have mentioned the Cassini and Catalan identities, but I wanted to do this without resorting to matrices (I don't know if it's possible)
Let, ${u_n}^2-{u_{n-1}}{u_{n+1}}=(-1)^{n-1}$.
Then,
$${u_{n+1}}^2-{u_{n}}{u_{n+2}}={u_{n+1}}^2-{u_{n}}(u_{n+1}+u_{n})$$
$$={u_{n+1}}(u_{n+1}-u_{n})-{u_n}^2={u_{n+1}}u_{n-1}-{u_n}^2=(-1)(-1)^{n-1}=(-1)^{(n+1)-1}$$
And this way, it is proved