IIT JEE past question AP GP 3

If a, b, c, are in AP.

a², b², c² are in HP

prove that either a=b=c or

a,b,-c/2 are in GP

1 Answers

a,b,c are in A.P
b-a=c-b

a²,b²,c² are in H.P

1/b²-1/a²=1/c²-1/b²

(a-b)(a+b)/a²b²=(b-c)(b+c)/b²c²

a+b/a²=b+c/c² (since a-b=b-c)

b(1/a²-1/c²)+(1/a-1/c)=0

(1/a-1/c)[b(1/a+1/c)+1]=0

either 1/a-1/c=0 a=c=b

or b(1/a+1/c)+1=0

b(a+c)=-ac

b²=-ac/2 a,b,-c/2 are in GP

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