HMMT 十一月 2016 · GEN 赛 · 第 5 题

HMMT November 2016 — GEN Round — Problem 5

Let the sequence { a } be defined by a = and a = 1 + ( a − 1) . Find the product i 0 n n − 1 i =0 2 ∞ ∏ a = a a a . . . i 0 1 2 i =0

解析

Let the sequence { a } be defined by a = and a = 1 + ( a − 1) . Find the product i 0 n n − 1 i =0 2 ∞ ∏ a = a a a . . . i 0 1 2 i =0 Proposed by: Henrik Boecken 2 Answer: 3 ∞ 2 Let { b } be defined by b = a − 1 and note that b = b . The infinite product is then i n n n i =0 n − 1 k 2 4 2 (1 + b )(1 + b )(1 + b ) . . . (1 + b ) . . . 0 0 0 0 By the polynomial identity k 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