# induction with 2 recursive sequnces

I'm having trouble solving this problem. I have relation for two sequences of natural numbers.

$$a(n)+\sqrt3\cdot b(n)=\left(1+\sqrt3\right)^n$$

and I have to prove that recursions:

\begin{align*} &a(n+1)=a(n)+3b(n)\\ &b(n+1)=a(n)+b(n) \end{align*}

will satisfy relation mentioned above. I tried solve it using induction but I failed. Can anybody help with some other ideas?

Induction works fine. If

$$a(n)+\sqrt3\cdot b(n)=\left(1+\sqrt3\right)^n\;,$$

then

\begin{align*} \left(1+\sqrt3\right)^{n+1}&=\left(1+\sqrt3\right)\big(a(n)+\sqrt3\cdot b(n)\big)\\ &=a(n)+3b(n)+\sqrt3\big(a(n)+b(n)\big)\;, \end{align*}

and the last expression is equal to $a(n+1)+\sqrt3\cdot b(n+1)$ if if we set

$$a(n+1)=a(n)+3b(n)$$

and

$$b(n+1)=a(n)+b(n)\;.$$