HMMT 二月 2007 · 冲刺赛 · 第 29 题

HMMT February 2007 — Guts Round — Problem 29

[ 15 ] A sequence { a } of positive reals is defined by the rule a a = a a for integers n > 2 n n ≥ 1 n +1 n − 1 n n − 2 together with the initial values a = 8 and a = 64 and a = 1024. Compute 1 2 3 √ √ √ a + a + a + · · · 1 2 3

解析

[ 15 ] A sequence { a } of positive reals is defined by the rule a a = a a for integers n > 2 n n ≥ 1 n +1 n − 1 n n − 2 together with the initial values a = 8 and a = 64 and a = 1024. Compute 1 2 3 √ √ √ a + a + a + · · · 1 2 3 √ Answer: 3 2 . Taking the base-2 log of the sequence { a } converts the multiplicative rule to a more n familiar additive rule: log ( a ) − 4 log ( a ) + 5 log ( a ) − 2 log ( a ) = 0. The characteristic n +1 n n − 1 n − 2 2 2 2 2 3 2 2 n equation is 0 = x − 4 x + 5 x − 2 = ( x − 1) ( x − 2), so log ( a ) is of the form a · n + b + c · 2 and we n 2 n − 1 2 n +2 find a = 2 . Now, n √ √ √ √ √ √ √ a + a + a + · · · = 2 · 4 + 16 + 64 + · · · . 1 2 3 We can estimate the new nested radical expression as 3, which expands thus √ √ √ √ √ √ 3 = 4 + 5 = 4 + 16 + 9 = 4 + 16 + 64 + 17 = · · · √ k k k +1 As a rigorous confirmation, we have 2 + 1 = 4 + (2 + 1), as desired. It follows that the answer √ is 3 2 .