Given that: $a_0=1,\ \ b_0=\dfrac{1}{\sqrt{2}}, \ \ t_0=\dfrac{1}{4}, \ \ p_0=1$
Define $a_{n+1}=\dfrac{a_n+b_n}{2} \ \ , b_{n+1}=\sqrt{a_n b_n} \\ t_{n+1}=t_n-p_n(a_n-a_{n+1})^2 \ \ , p_{n+1}=2p_n.$
Find the limit:
$$\lim_{n \to \infty} \frac{(a_n+b_n)^2}{4t_n}$$
I tried to solve the question but I couldn't find a way to start, so I calculated the values up to $10000$ and it seems to be approaching $\pi$. How do I prove it?