Problem 10: Sequence & inequality

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by henderson, May 4, 2016, 8:49 PM

$\bf\color{red}Problem \ 10 \$ $m_1<m_2<...<m_k$ are positive integers, such that $\dfrac{1}{m_1}, \dfrac{1}{m_2} ,...,\dfrac{1}{m_k}$ are in arithmetic progression. Prove that $k<m_1+2.$
$\bf\color{red}Solution \$ Let $d=\frac{1}{m_1}-\frac{1}{m_2}\ge \frac{1}{m_1m_2}.$
So
$\frac{1}{m_2}>\sum_{i=2}^{k-1}(\frac{1}{m_{i}}-\frac{1}{m_{i+1}})=(k-2)d\ge \frac{k-2}{m_1m_2},$ which implies $m_1>k-2.$ $\square$

This post has been edited 9 times. Last edited by henderson, Feb 5, 2017, 8:40 AM

Math Path

Problem 10: Sequence & inequality

by henderson, May 4, 2016, 8:49 PM

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