a-b-c-are-in-AP-or-GP-or-HP-according-as-a-b-b-c-is-equal-to-

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}