Answer :

Let the two numbers be ‘a’ and ‘b’

The arithmetic mean is given by and the geometric mean is given by

We have to insert two geometric means between a and b

Now that we have the terms a, G_{1}, G_{2}, b

G_{1} will be the geometric mean of a and G_{2} and G_{2} will be the geometric mean of G_{1} and b

Hence and

Square

⇒ G_{1}^{2} = aG_{2}

Put

Square both sides

⇒ G_{1}^{4} = a^{2}(G_{1}b)

⇒ G_{1}^{3} = a^{2}b

Put value of G_{1} in

Now we have to prove that

Consider RHS

Substitute values of G_{1} and G_{2} from (i) and (ii)

⇒ RHS = a + b

Divide and multiply by 2

But

Hence

⇒ RHS = 2A

Hence RHS = LHS

Hence proved

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