Concept:

Let the two numbers as a and b.

A.M. = \(\rm \dfrac {a+b}{2}\)

G.M. = \(\rm\sqrt{ab}\) 

Relation between AM and GM:

(G.M.)2 = (A.M.)(H.M.)

Calculations:

Given, the arithmetic mean of two numbers exceeds their geometric mean by 2 and the geometric mean exceeds their harmonic mean by 1.6.

⇒ A.M. = G.M. + 2         ....(1)

and G.M. = H.M. + 1.6

We know that, (G.M.)² = (A.M.)(H.M.)

⇒ (G.M.)2 = (G.M. + 2)(G.M. - 1.6)

⇒ (G.M.)2 = (G.M.)² + 0.4 G.M. - 3.2 

⇒ 0.4 G.M. = 3.2

⇒ G.M. = 8                  ....(2)

From equation (1), we get 

⇒A.M. = 10                  ....(3)

Consider, the two numbers as a and b.

then, G.M. = \(\rm\sqrt{ab}\) and A.M. = \(\rm \dfrac {a+b}{2}\)

From equation (2) and (3), we get

⇒ \(\rm\sqrt{ab}\) = 8 and \(\rm \dfrac {a+b}{2}\) = 10

⇒ ab = 64 and a + b = 20 

⇒ a (20 - a) = 64

⇒ a²  - 20a - 64 = 0

⇒ a =16 or a = 4

⇒ b = 4 or b = 16

Hence, the two numbers are 16 and 4.

Hence, The arithmetic mean of two numbers exceeds their geometric mean by 2 and the geometric mean exceeds their harmonic mean by 1.6. the two numbers are 16 and 4.