Convert a recursive formula to explicit

Question

Given sequence:

$A_{1} = 0$ $A_{2} = 3$ $A_{3} = \frac{3}7$ $A_{4} = \frac{21}{13}$ $A_{5} = \frac{39}{55}$

And the recursive formula is given as $_{n+1}=\frac{3}{2_{n} + 1}$

How can one find this sequence's explicit formula?

Answer 1

Hint: Solve $x = \frac{3}{2 x + 1}$. Why is this a good approach?

Alternatively, consider the subsequences of $A_n$ for odd $n$ and for even $n$ and prove that they are monotonic and bounded.

Answer 2

It's a Möbius transformation.

check my answer at Recursive sequence depending on the parameter for a shortcut method. A few others have also posted such approach in the past.