QOD-28/05/13

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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

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Debargha Paul ·2013-05-31 00:43:00

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=âˆšac
we know 2b=a+c or,4b²=a²+c²+2ac
or,4ac=a²+c²+2ac
or(a-c)²=0
so a=c
bâˆšac
or,b=a
so a=b=c

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QOD-28/05/13

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