Can there be more than 5 positive integers such that they are in a harmonic progression ie there reciprocals are in arithmetic progression? If yes, the find some with more than 5 terms and show your working, and if not, prove your observation.

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TopNewestNote that \(\frac{1}{n!}, \frac{2}{n!}, \frac{3}{n!}, \ldots, \frac{n}{n!}\) form an arithmetic progression, and the numerator divides the denominator so they all simplify to unit fractions. Thus their reciprocals are positive integers that form a harmonic progression. Take \(n\) as large as you want. – Ivan Koswara · 8 months, 3 weeks ago

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– Kushagra Sahni · 8 months, 3 weeks ago

That's a cool solution. Do other solutions exist?Log in to reply

Whether the above gives all the solutions, I haven't proved it yet, although I think it does. – Ivan Koswara · 8 months, 3 weeks ago

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– Kushagra Sahni · 8 months, 3 weeks ago

Yes I guess that covers all.Log in to reply