1
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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

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

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.