If one A.M., A and two geometric means G₁ and G₂ inserted between any two positive numbers, show that

Question

If one A.M., A and two geometric means G 1 and G 2 inserted between any two positive numbers, show that

Philoid · Accepted Answer

Let the numbers be a and b.

Now or 2A =a+b

Also, G₁ and G₂ are GM between a and b, then a, G₁, G₂, b are in G.P.

Let r be the common ratio.

Then, b = ar^4–1 = ar³

⇒

∴ G₁ = ar =

G₂ = ar² =

∴

a + b = 2A

If one A.M., A and two geometric means G1 and G2 inserted between any two positive numbers, show that