2003 Putnam B2

Let $n$ be a positive integer. Starting with the sequence

$1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n}$,

form a new sequence of $n-1$ entries

$\frac{3}{4}, \frac{5}{12}, \dots, \frac{2n-1}{2n(n-1)}$

by taking the averages of

two consecutive entries in the first sequence. Repeat the

averaging of neighbors on the second sequence to obtain a third

sequence of $n-2$ entries, and continue until the final sequence produced

consists of a single number $x_n$. Show that $x_n < 2/n$.

令 $n$ 为正整数。从顺序开始

$1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n}$,

形成 $n-1$ 条目的新序列

$\frac{3}{4}、\frac{5}{12}、\dots、\frac{2n-1}{2n(n-1)}$

通过取平均值

第一个序列中的两个连续条目。重复

averaging of neighbors on the second sequence to obtain a third

$n-2$ 条目的序列，并继续直到生成最终序列

由单个数字 $x_n$ 组成。表明 $x_n < 2/n$。