Today, I was handing a lot with Progressions. It came to my mind that there are infinite AP and GP. for example :- An Arithmetic Progression with First term 2 and common difference 5 looks like the following :- 2,7,12,17 ...... and it bursts to infinity
Similarly, An Geometric Progression with First term 2 and common ratio 5 looks like the following :- 2,10,50,250.... and this also burst to infinity (By infinity, I mean all positive integers)
But then what about Harmonic Progressions, do they also bursts to infinity?
I think that there doesn't exists any infinite Harmonic Progression containing all positive integers.
(Insight is that it's the reciprocal of Arithmetic Progression)
But How do I prove it?
(I tried a lot but couldn't get a final result.)
Your help will be highly appreciated.