QOD-28/05/13

a,b,c are in AP & a2,b2,c2 are in HP, then prove that a=b=c or a,b,-c/2 are in GP

1 Answers

11
Debargha Paul ·

SINCE a2,b2,c2 ARE IN HP
SO b2=2a2c2/a2+c2
Again 2b=a+c
or,a2+c2=4b2-2ac

so 4b4-2ab2ac=2a2c2
or,(4b2+2ac)(b2-ac)=0
either 4b2=-2ac
or,b2=-ac/2
so a,b,-c/2 are in gp
OR,b=√ac
we know 2b=a+c or,4b2=a2+c2+2ac
or,4ac=a2+c2+2ac
or(a-c)2=0
so a=c
b√ac
or,b=a
so a=b=c

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