Question 1185523


I will do the first progression only-- insert 3 harmonic means between 1/2 and 1/6.

If five numbers are in harmonic progression then their reciprocals are in arithmetic progression.  

Let those numbers in AP be 2, a, b, c, 6.

Then it's quite clear that the numbers in between are 2, {{{red(3)}}},{{{ red(4)}}}, {{{red(5)}}}, 6.

Therefore the three harmonic means between 1/2 and 1/6 are 1/3, 1/4, and 1/5.