MathDB

4

Part of 1973 Dutch Mathematical Olympiad

Problems(1)

x_{n+1} = \sqrt{x_n -\frac14}

Source: Netherlands - Dutch NMO 1973 p4

1/27/2023
We have an infinite sequence of real numbers x0,x1,x2,...x_0,x_1, x_2, ... such that xn+1=xn−14x_{n+1} = \sqrt{x_n -\frac14} holds for all natural nn and moreover x0∈12x_0 \in \frac12. (a) Prove that for every natural nn holds: xn>12x_n > \frac12 (b) Prove that lim⁡n→∞xn\lim_{n \to \infty} x_n exists. Calculate this limit.
recurrence relationSequencealgebralimit