How Might Geometry Be Useful Here?

Calculus Level 3

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

s0=2;sn+1=24sn2s_0 = 2; s_{n+1} = \sqrt{2-\sqrt{4-s_n^2}} for n0n \ge 0.

To the nearest hundredth, find the value of limn2nsn\displaystyle\lim_{n \to \infty} 2^n s_n.

In other words, to what value does the following sequence converge:23s3=822+22^3 s_3 = 8\sqrt{2-\sqrt{2+\sqrt{2}}}24s4=1622+2+22^4 s_4 = 16\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}25s5=3222+2+2+22^5 s_5 = 32\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}and so on...

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