# How Might Geometry Be Useful Here?

Calculus Level 2

Let $(s_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows:

$s_0 = 2; s_{n+1} = \sqrt{2-\sqrt{4-s_n^2}}$ for $n \ge 0$.

To the nearest hundredth, find the value of $\displaystyle\lim_{n \to \infty} 2^n s_n$.

In other words, to what value does the following sequence converge:$2^3 s_3 = 8\sqrt{2-\sqrt{2+\sqrt{2}}}$$2^4 s_4 = 16\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}$$2^5 s_5 = 32\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}$and so on...

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