Let $$m_1 < m_2 < \dots <m_{k − 1} < m_k$$ be distinct positive integers such that their reciprocals are in arithmetic progression.
- Show that $k < m_1+2$.
- Give an example of such a sequence of length $k$ for any positive integer $k$.
My approach : $$\frac{1}{m_k}=\frac{1}{m_1}+(k-1)d \\ \frac{m_1-m_k}{m_1m_kd}=k-1$$ So $d$ must be a fraction. Am I going right?
As regards the first part see Integer reciprocals in arithmetic progression .
For the second part, given a positive integer $k$, you may try the integer sequence $$\frac{k!}{k}<\frac{k!}{k-1}<\dots<\frac{k!}{3}<\frac{k!}{2}<k!$$ then the difference between the reciprocals of two consecutive terms is $1/k!$.