HMMT 二月 2007 · 代数 · 第 7 题

HMMT February 2007 — Algebra — Problem 7

[ 5 ] An infinite sequence of positive real numbers is defined by a = 1 and a = 6 a − a for 0 n +2 n n +1 n = 0 , 1 , 2 , · · · Find the possible value(s) of a . 2007

解析

[ 5 ] An infinite sequence of positive real numbers is defined by a = 1 and a = 6 a − a for 0 n +2 n n +1 n = 0 , 1 , 2 , · · · Find the possible value(s) of a . 2007 2007 2 Answer: 2 . The characteristic equation of the linear homogeneous equation is m + m − 6 = ( m + 3)( m − 2) = 0 with solutions m = − 3 and m = 2. Hence the general solution is given by n n a = A (2) + B ( − 3) where A and B are constants to be determined. Then we have a > 0 for n ≥ 0, n n n so necessarily B = 0, and a = 1 ⇒ A = 1. Therefore, the unique solution to the recurrence is a = 2 0 n for all n.