Induction proof for continued fractions

Question

Recently while preparing for a maths test, I got this question in a book:

Let $a(n) = 3 + \cfrac{1}{3+\cfrac{1}{3+\cfrac{1}{3+\cdots }}}$ till $n$ terms.

Prove that $a(n) \cdot a(n-1)=3a(n-1)+1$ for $n \geqslant 2$.

I just don't know how to start proving that the equation is valid for $(k+1)$th term also.

Any help would be surely appreciated. :) Thanks in advance.

Answer 1

HINT

$a(n) = 3 + \dfrac{1}{\color{red}{3+\dfrac{1}{3+\dfrac{1}{3+\cdots }}}} = 3 + \dfrac{1}{\color{red}{a(n-1)}} $