Solution:

Given, Arithmetic mean (A.M) and Geometric mean (G.M) = 15

Let the 10 observations be a1, a2, a3, a4, a5, a6, a7, a8, a9, a10

Arithmetic Mean (A.M) = (a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10)/10 = 15

Geometric Mean (G.M) = (a1 x a2 x a3 x a4 x a5 x a6 x a7 x a8 x a9 x a10)^(1/10) = 15

To find Harmonic Mean (H.M), we use the formula:

H.M = n/(1/a1 + 1/a2 + 1/a3 + 1/a4 + 1/a5 + 1/a6 + 1/a7 + 1/a8 + 1/a9 + 1/a10)

where, n = number of observations

Substituting the given values, we get:

H.M = 10/(1/a1 + 1/a2 + 1/a3 + 1/a4 + 1/a5 + 1/a6 + 1/a7 + 1/a8 + 1/a9 + 1/a10)

We need to find a1, a2, a3, a4, a5, a6, a7, a8, a9, a10 to find H.M.

Let's assume a1 = a2 = a3 = a4 = a5 = a6 = a7 = a8 = a9 = a10 = x

Substituting this in A.M formula, we get:

A.M = (10x)/10 = x

x = 15

Substituting this in G.M formula, we get:

G.M = (15^10)^(1/10) = 15

Substituting this in H.M formula, we get:

H.M = 10/(1/15 + 1/15 + 1/15 + 1/15 + 1/15 + 1/15 + 1/15 + 1/15 + 1/15 + 1/15)

H.M = 10/(10/15)

H.M = 15

Therefore, Harmonic Mean (H.M) = 15.

AM*HM=GM^2
15*HM=(15)^2
HM=225/15
HM=15