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

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


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

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




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