I would like to share a simple derivation of an iterative formula for Pi.
The formula I have derived: $\begin{align} A_1 &= 4 \\ A_{n+1} &=2*4^{n} \left(1 - \sqrt{1 - \frac{A_n}{4^{n}}}\right). \\ \lim_{n\to\infty} A_n &= \pi^2 \end{align}$
Note: After a bit of research I learned that formula above is closely related to relations discovered a long time ago by Euler.
Despite the fact that such formula isn’t very special, I believe that together with the derivation below, it will provide a simple explanation for how Pi can be calculated using simple tools. (Without calculus for example)
Will be happy to hear feedback!
An informal derivation of the formula above:
