a, b, c are in AP or GP or HP according as ((a−b)/(b−c)) is equal to


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Question Number 46923 by 786786AM last updated on 02/Nov/18

$a, b, c are in AP or GP or HP according$ $as \frac{a - b}{b - c} is equal to$
	Answered by tanmay.chaudhury50@gmail.com last updated on 02/Nov/18

	$1) w h e n a, b, c a r e i n A . P$ $t h e n a - b = b - c s o \frac{a - b}{b - c} = 1$ $2) w h e n a, b, c a r e i n G P$ $b = a r c = {a r}^{2}$ $\frac{a - b}{b - c} = \frac{a - a r}{a r - {a r}^{2}} = \frac{a (1 - r)}{a r (1 - r)} = \frac{1}{r} = r e c i p r o c a l o f$ $c o m m o n r a t i o$ $3) w h e n a, b, c a r e i n G P$ $\frac{2}{b} = \frac{1}{a} + \frac{1}{c}$ $\frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b}$ $\frac{a - b}{a b} = \frac{b - c}{b c}$ $\frac{a - b}{b - c} = \frac{a}{c}$
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