HMMT 二月 2020 · 冲刺赛 · 第 27 题

HMMT February 2020 — Guts Round — Problem 27

[15] Let { a } be a sequence of real numbers defined by i i ≥ 0 1 2 a = a − n +1 n n 2020 · 2 − 1 2 for n ≥ 0. Determine the largest value for a such that { a } is bounded. 0 i i ≥ 0

解析

[15] Let { a } be a sequence of real numbers defined by i i ≥ 0 1 2 a = a − n +1 n n 2020 · 2 − 1 2 for n ≥ 0. Determine the largest value for a such that { a } is bounded. 0 i i ≥ 0 Proposed by: Joshua Lee 1 Answer: 1 + 2020 2 ( ) 1 1 1 √ √ Solution: Let a = t + , with t ≥ 1. (If a < then no real t exists, but we ignore 2020 2018 0 0 t 2 2 these values because a is smaller.) Then, we can prove by induction that 0 ( ) n 1 1 2 a = t + . n n √ n 2020 · 2 2 t 2 For this to be bounded, it is easy to see that we just need n ( ) 2 n 2 t t = n √ √ 2020 · 2 2020 2 2 2020 / 2 to be bounded, since the second term approaches 0. We see that this is is equivalent to t ≤ 2 , which means ( ) ( ) 2020 √ 2020 1 1 1 a ≤ 2 + √ = 1 + . 0 √ 2020 2020 2 2 2