
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 $\displaystyle u_{n+1} = Pu_n  Qu_{n1}$ with $\displaystyle u_0 = 0, u_1 = 1$.
Similar define $\displaystyle v_{n+1} = Pv_n  Qv_{n1}$ with $\displaystyle v_0 = 2, v_1 = P$.
Show:
(1) $\displaystyle 2u_n = Pu_{n1} + v_{n1}$ (Managed to prove by induction myself)
(2) $\displaystyle 2v_n = (P^24Q)u_{n1} + Pv_{n1}$
(3) $\displaystyle u_{2n} = u_{n}v_{n}$
(4) $\displaystyle v_{2n} = v_{n}^2  2Q^{n}$
(5) $\displaystyle u_{n+m} = u_{n}u_{m+1} Qu_{m}u_{n1} = (u_{n}v_{m} + u_{m}v_{n})/2$
(6) $\displaystyle v_{n+m} = v_{n}v_{m} Q^{m}v_{nm}$
I would be thankful for any proven statement :).

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