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Putnam 2003 B2

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June 23, 2011
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Problem Statement

Let nn be a positive integer. Starting with the sequence 1,12,13,⋯ ,1n1,\frac{1}{2}, \frac{1}{3} , \cdots , \frac{1}{n}, form a new sequence of n−1n -1 entries 34,512,⋯ ,2n−12n(n−1)\frac{3}{4}, \frac{5}{12},\cdots ,\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−2n -2 entries and continue until the final sequence consists of a single number xnx_n. Show that xn<2nx_n < \frac{2}{n}.
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