Tag Archives: sequence

2013-17 Repeated numbers

A real sequence \( x_1, x_2, x_3, \cdots \) satisfies the relation \( x_{n+2} = x_{n+1} + x_n \) for \( n = 1, 2, 3, \cdots \). If a number \( r \) satisfies \( x_i = x_j = r \) for some \( i \) and \( j \) \( (i \neq j) \), we say that \( r \) is a repeated number in this sequence. Prove that there can be more than \( 2013 \) repeated numbers in such a sequence, but it is impossible to have infinitely many repeated numbers.

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2013-09 Inequality for a sequence

Let \( N > 1000 \) be an integer. Define a sequence \( A_n \) by
\[
A_0 = 1, \, A_1 = 0, \, A_{2k+1} = \frac{2k}{2k+1} A_{2k} + \frac{1}{2k+1} A_{2k-1}, \, A_{2k} = \frac{2k-1}{2k} \frac{A_{2k-1}}{N} + \frac{1}{2k} A_{2k-2}.
\]
Show that the following inequality holds for any integer \( k \) with \( 1 \leq k \leq (1/2) N^{1/3} \).
\[
A_{2k-2} \leq \frac{1}{\sqrt{(2k-2)!}}.
\]

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