1. ## Lucas Sequences

I read some facts about Lucas sequences, but they were not proven, can you help me?

Let P and Q be integers. Define $u_{n+1} = Pu_n - Qu_{n-1}$ with $u_0 = 0, u_1 = 1$.
Similar define $v_{n+1} = Pv_n - Qv_{n-1}$ with $v_0 = 2, v_1 = P$.

Show:
(1) $2u_n = Pu_{n-1} + v_{n-1}$ (Managed to prove by induction myself)
(2) $2v_n = (P^2-4Q)u_{n-1} + Pv_{n-1}$
(3) $u_{2n} = u_{n}v_{n}$
(4) $v_{2n} = v_{n}^2 - 2Q^{n}$
(5) $u_{n+m} = u_{n}u_{m+1} -Qu_{m}u_{n-1} = (u_{n}v_{m} + u_{m}v_{n})/2$
(6) $v_{n+m} = v_{n}v_{m} -Q^{m}v_{n-m}$

I would be thankful for any proven statement .

2. If you assume $n \leq m$ you could prove (5) and (6) by induction where your base cases are when $n=m$, which are (3) and (4) respectively.