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Question

If between any two quantities there be inserted two arithmetic means A1,A2; two geometric means G1,G2:H1H2=A1+A2:H1+H2.

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Solution

Let the quantities be a and b. Then, a,A1,A2,b are in a.p. Thus, we can write

A1a=bA2 or A1+A2=a+b(I)

If a,G1,G2 are in G.P.Then we can write

G1a=bG2 or G1G2=ab(II)

If a,H1,H2,b are in HP. Then 1a,1H1,1H2,1b will be in A.P.

1H11a=1b1H2 or,

1H1+1H2=1a+1b=a+bab

From (I) and (II),

1H1+1H2=A1+A2G1G2


H1+H2H1H2=A1+A2G1G2

G1G2H1H2=A1+A2H1+H2

G1G2:H1H2=(A1+A2):(H1+H2)

Hence proved


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