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Question

If A1,A2; G1,G2 and H1,H2 are arithmetic mean, geometric mean and harmonic mean between two numbers, then the value of G1G2H1H2×H1+H2A1+A2 is

A
1
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B
0
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C
2
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D
3
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Solution

The correct option is A 1
Let a and b be two numbers.
Sum of n A.M.'s=n×single A.M.
A1+A2=2×(a+b2)=a+b (1)

Product of n G.M.'s=(single G.M.)n
G1G2=(ab)2=ab (2)

Dividing equation (1) by (2), we get
A1+A2G1G2=a+bab (3)
Since a,H1,H2,b are in H.P.
1a,1H1,1H2,1b are in A.P.
1H1+1H2=2⎜ ⎜ ⎜1a+1b2⎟ ⎟ ⎟
1H1+1H2=1a+1b=a+bab
H1+H2H1H2=a+bab (4)

From (3) and (4), we get
A1+A2G1G2=H1+H2H1H2
H1H2H1+H2=G1G2A1+A2
G1G2H1H2×H1+H2A1+A2=1

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