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Question

If a,b,c are in A.P., and a2,b2,c2 are in H.P., then-

A
a=b=c
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B
a2=b2=c22
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C
a,b,c are in G.P.
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D
a2,b,c are in G.P.
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Solution

The correct options are A a=b=c C a,b,c are in G.P. D −a2,b,c are in G.P.We have 2b=a+c and b2=2a2c2a2+c2.....(i)On eliminating b, we get8a2c2=(a2+c2+2ac)(a2+c2)Which can be arranged as(a2+c2−2ac)(a2+c2+4ac)=0⇒either a=c or (a+c)2+2ac=0If a=c then a=b=c⇒ a, b, c may be treated as three numbers in G .P.If (a+c)2+2ac=0, then by using (i) choice (D) follows.

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