
Let a, b and c form an H.P. Then 1/a, 1/b and 1/c form an A.P.

If a, b and c are in H.P. then 2/b = 1/a + 1/c, which can be simplified as b = 2ac/(a+c)

If a and b are two nonzero numbers then the sequence a, H, b is a H.P.
 The n numbers H1, H2, ……,Hn are said to be harmonic means between a and b,
 if a, H1, H2 ……, Hn, b are in H.P. i.e. if 1/a, 1/H1, 1/H2, ..., 1/Hn, 1/b are in A.P.
 Let d be the common difference of the A.P., Then 1/b = 1/a + (n+1) d ⇒ d = a–b/(n+1)ab.

If a andb are two positive real numbers then A.M x H.M = G.M2

The relation between the three means is defined as A.M > G.M > H.M

If we need to find three numbers in a H.P. then they should be assumed as 1/a–d, 1/a, 1/a+d

Four convenient numbers in H.P. are 1/a–3d, 1/a–d, 1/a+d, 1/a+3d

Five convenient numbers in H.P. are 1/a–2d, 1/a–d, 1/a, 1/a+d, 1/a+2d