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Trigonometric Sequence

Source: 1990 IrMO Paper 2 Problem 3

September 30, 2017
trigonometry

Problem Statement

Let tt be a real number, and let a_n=2\cos \left(\frac{t}{2^n}\right)-1,  n=1,2,3,\dots Let bnb_n be the product a1a2a3⋯ana_1a_2a_3\cdots a_n. Find a formula for bnb_n that does not involve a product of nn terms, and deduce that lim⁡n→∞bn=2cos⁡t+13\lim_{n\to \infty}b_n=\frac{2\cos t+1}{3}
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