HMMT 二月 2012 · 代数 · 第 6 题

HMMT February 2012 — Algebra — Problem 6

Let a = − 2, b = 1, and for n ≥ 0, let 0 0 √ 2 2 a = a + b + a + b , n +1 n n n n √ 2 2 b = a + b − a + b . n +1 n n n n Find a . 2012

解析

Let a = − 2, b = 1, and for n ≥ 0, let 0 0 √ 2 2 a = a + b + a + b , n +1 n n n n √ 2 2 b = a + b − a + b . n +1 n n n n Find a . 2012 √ 1006 2011 2010 Answer: 2 2 + 2 − 2 We have a + b = 2( a + b ) n +1 n +1 n n 2 2 2 a b = ( a + b ) − a − b = 2 a b n +1 n +1 n n n n n n Thus, n a + b = − 2 n n n +1 a b = − 2 n n Using Viete’s formula, a and b are the roots of the following quadratic, and, since the square 2012 2012 root is positive, a is the bigger root: 2012 2 2012 2013 x + 2 x − 2 Thus, √ 1006 2011 2010 a = 2 2 + 2 − 2 2012