Let p,p1 be A.M. and G.M. between a and b respectively and q,q1 be the A.M. and G.M. between b and c respectively where a,b,c>0. If a,b,c are in A.P., then which of the following is CORRECT?
A
p2−q2=q21−p21
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B
p2−q2=p21−q21
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C
p2+q2=p21−q21
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D
p2−q2=p21+q21
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Solution
The correct option is Bp2−q2=p21−q21 Given : p=a+b2,q=b+c2p1=√ab,q1=√bc
a,b,c are in A.P. ⇒2b=a+c Now, p2−q2=14(a2+b2+2ab−b2−c2−2bc)⇒p2−q2=14(a2−c2+2b(a−c))⇒p2−q2=(a−c)(a+c+2b)4⇒p2−q2=(a−c)(2b+2b)4⇒p2−q2=b(a−c)⇒p2−q2=ab−bc∴p2−q2=p21−q21