Q.

if A is the arithmetic mean and G1, G2 be two geometric means between any two numbers, then G12G2+G22G1 is equal to

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a

2A

b

A

c

3A

d

None of these

answer is A.

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Detailed Solution

Let A be the  arithmetic  mean  between  two numbers a and b, Again, let G1,andG2,  be  the two geometric mean between  two  numbers a  and b.  ∴ A=a+b2 and G1=ar,G2=ar2where r=ba12+1=ba1/3   ……(i)∴ G12G2+G22G1=a2r2ar2+a2r4ar=a+ar3=a+aba [∵ from Eq. (i) ]=a+b=2A ∵A=a+b2
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if A is the arithmetic mean and G1, G2 be two geometric means between any two numbers, then G12G2+G22G1 is equal to