How Might Geometry Be Useful Here?

Calculus Level 4

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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