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If a, b, c be in A.P. and a 2, b 2 , c 2 in H.P. then prove that either -a/2, b, c are in GP or a=b=c.

If a, b, c be in A.P. and a2, b2, c2 in H.P. then prove that either -a/2, b, c are in GP or a=b=c.

Grade:12

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
6 years ago
Hello student,
Please find the answer to your question below
According to the problem
b=(a+c)/2......(1) and b2=(2a2c2)(a2+c2).........(2)
From 1 (a+c)2=4b2
So (a+c)2=8a2c2/(a2+c2)
on expanding and simplifying the above equation we get
(a2+c2)2+2ac(a2+c2)-8a2c2=0
(a2-c2)2+2ac(a2+c2-2ac)=0
(a2-c2)2+2ac(a-c)2=0
So (a-c)2[a2+c2+4ac]=0
i.e.,a=c or a2+c2=-4ac
if a=c then b=(a+c)/2=a=c
So a=b=c or b2=(2a2c2)/-4ac=-ac/2
i.e., -a,b,-c/2 are in GP

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