HMMT 十一月 2016 · GEN 赛 · 第 5 题

英文解析

5. Let the sequence { a } be defined by a = and a = 1 + ( a − 1) . Find the producti 0 n n − 1
   i =0
   ∞2
   ∏
   a = a a a . . .
   i 0 1 2
   i =0
   Proposed by: Henrik Boecken
   Answer:2
   ∞ 23
   Let { b } be defined by b = a − 1 and note that b = b . The infinite product is theni n n ni =0 n − 1
   2 4 2 k
   (1 + b )(1 + b )(1 + b ) . . . (1 + b ) . . .
   0 0 00
   By the polynomial identityk 1
   2 4 2 2 3
   (1 + x )(1 + x )(1 + x ) . . . (1 + x ) · · · = 1 + x + x + x + · · · =
   1 − x
   Our desired product is then simply
   1 2
   1 − ( a − 1) 3= 0