calculus - Is there a general form for $a_n=sqrt{2+a_{n-1}}$?

Can I find the general term of this sequence $a_n=\sqrt{2+a_{n-1}}$, $a_1=\sqrt2$? I have proved the convergence. And found its limit. But is there any general form for it?

Answer

Hint: If $a_n=2 \cos(\theta_n)$ with $0 < \theta_n<\pi/2$, then:

$\sqrt{2+a_n}=\sqrt{2+2 \cos(\theta_n)}$

$=2 \sqrt{(1+\cos(\theta_n))/2}$

$=2 \cos((\theta_n)/2)$ using the appropriate half-angle formula.