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# Nested Roots

Consider the set $$S_n$$ of all the $$2^n$$ numbers of the type $$2\pm\sqrt{2\pm\sqrt{2\pm\dots}}$$, where the number $$2$$ appears $$n+1$$ times.

(a) Show that all members of $$S_n$$ are real.

(b) Find the product $$P_n$$ of all elements of $$S_n$$.

Source: Austria 1989

Note by Cody Johnson
2 years, 3 months ago

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(a)Assume, up to a number n, that all values are positive and real.These values are bounded by 4, since the infinite radical $$\sqrt{2+\sqrt{2+...}}=2$$.So the n+1st number is bigger than $$2-\sqrt{2+\sqrt{2+...}}=0$$.Thus, by induction, every such number is real and positive (the base case is obvious)

(b)I will show that the product is the same for every n.Let $$\alpha_n$$ be a number $$\sqrt{2\pm\sqrt{2\pm...}}$$.That means that$$P_n=\displaystyle\prod_{}{}(2+\alpha_n)(2-\alpha_n)=\prod 4-\alpha_n^2=\prod (2+\alpha_{n-1})(2+\alpha_{n-1})=P_{n-1}$$.Since $$P_1=2$$, they are all 2. · 2 years, 3 months ago

Bravo ! · 2 years, 3 months ago

Thanks :D .By the way when is the next Proofathon? · 2 years, 3 months ago