If A is the arithmetic mean and G₁, G₂ be two geometric means between any two numbers, then prove that

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, G₁, G₂, b

G₁ will be the geometric mean of a and G₂ and G₂ will be the geometric mean of G₁ and b

Hence and

Square

⇒ G₁² = aG₂

Put

Square both sides

⇒ G₁⁴ = a²(G₁b)

⇒ G₁³ = a²b

Put value of G₁ in

Now we have to prove that

Consider RHS

Substitute values of G₁ and G₂ from (i) and (ii)

⇒ RHS = a + b

Divide and multiply by 2

But

Hence

⇒ RHS = 2A

Hence RHS = LHS

Hence proved