If a,b,c, are in A.P. and p, p' are respectively A.M. and G.M. between a and b while q, q' are respectively AM. And G.M. between b and c, then

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Solution

The correct option is C
We have 2b = a + c and a, p, b, q, c are in A.P
$\Rightarrow p = \frac{a + b}{2}, q = \frac{b + c}{2}$
Again, $p^{'} = \sqrt{a b} a n d q^{'} = \sqrt{b c}$
$∴ p^{2} - q^{2} = \frac{(a + b)^{2} - (b + c)^{2}}{4}$
$= \frac{(a - c) (a + c + 2 b)}{4} = (a - c) b = p^{^{'} 2} - q^{^{'} 2}$

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If a,b,c, are in A.P. and p, p' are respectively A.M. and G.M. between a and b while q, q' are respectively AM. And G.M. between b and c, then

The correct option is C We have 2b = a + c and a, p, b, q, c are in A.P ⇒p=a+b2,q=b+c2 Again,p′=√ab and q′=√bc ∴p2−q2=(a+b)2−(b+c)24 =(a−c)(a+c+2b)4=(a−c)b=p′2−q′2