Solution October 3, 2007

Problem

The sequence ${a n}$ is defined by $a_{0}= 2007,\, a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}$ for $n \ge 1$ . Prove that $\lfloor a_{n}\rfloor =2007-n$ for $0 \le n \le 1004$ , where $\lfloor x\rfloor$ denotes the largest integer no larger than x.

Solution

The recurrence is equivalent to

$a_{n+1} = a_n-1 + \frac1{a_n+1}$

$a_{k} = a_0 - k + \sum_{i=0}^{k-1} \frac1{a_i+1}$

Note that $a_{k} \geq a_0 - k = 2007 - k$ . Therefore we have

$\sum_{i=0}^{1003} \frac1{a_i+1} \leq \sum_{i=0}^{1003} \frac1{2008-i} < \sum_{i=0}^{1003} \frac1{1005} = \frac{1004}{1005} < 1$

Also, $\frac1{a_i+1}$ is positive for $i < 1004$ . Thus,

$0 \leq \sum_{i=0}^{k-1} \frac1{a_i+1} \leq \sum_{i=0}^{1003} \frac1{a_i+1} < 1$

for $0\leq k \leq 1004$ . Thus,

$2007 - k \leq a_{k} < 2008 - k$

which implies $2007 - k = \lfloor a_{k} \rfloor$ , completing the proof.

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