Problem #PRU-115397

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Problem

The sequence $a_{1}, a_{2}, \dots$ is such that $a_{1} \in (1, 2)$ and $a_{k + 1} = a_{k} + \frac{k}{a_{k}}$ for any positive integer $k$ . Prove that it cannot contain more than one pair of terms with an integer sum.The sequence $a_{1}, a_{2}, \dots$ is such that $a_{1} \in (1, 2)$ and $a_{k + 1} = a_{k} + \frac{k}{a_{k}}$ for any positive integer $k$ . Prove that it cannot contain more than one pair of terms with an integer sum.

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