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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
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Solution

## The correct option is A 1Let 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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