HMMT 二月 2007 · 冲刺赛 · 第 29 题
HMMT February 2007 — Guts Round — Problem 29
题目详情
英文原题
- [ 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
英文解析
- [ 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 morenfamiliar additive rule: log ( a ) − 4 log ( a ) + 5 log ( a ) − 2 log ( a ) = 0. The characteristicn +1 n n − 1 n − 2
2 2 2 2
3 2 2 nequation 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
2 nn − 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 +1 k
As a rigorous confirmation, we have 2 + 1 = 4 + (2 + 1), as desired. It follows that the answer
√
is 3 2 .