Question

If between two quantities, there be inserted two arithmetic means A1,A2
geometric means G1,G2, and two harmonic means H1,H2 then show that G1:G2
= ( A1 +A2): (H1 +H2).​

Answer 1

it is given that If between two quantities, there be inserted two arithmetic means A1,A2

geometric means G1,G2, and two harmonic means H1,H2.

we have to show that G1.G2/H1.H2 = (A1 + A2)/(H1 + H2)

let a, A1 , A2 , b are in Arithmetic progression.

A1 + A2 = a + b .........(1)

a, G1 , G2 , b are in geometric progression.

G1/a = G2/G1 = b/G2

⇒G1² = aG2 , G2² = bG1

⇒G1².G2² = abG1.G2

⇒G1.G2 = ab ..........(2)

a , H1 , H2 , b are in harmonic progression.

1/H1 - 1/a = 1/b - 1/H2

⇒1/H1 + 1/H2 = 1/b + 1/a

⇒(H1 + H2)/H1.H2 = (a + b)/ab

from equation (1) and (2) we get,

⇒(H1 + H2)/H1.H2 = (A1 + A2)/G1.G2

⇒G1.G2/H1.H2 = (A1 + A2)/(H1 + H2)