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Question -

Show that each one of the following progressions is a G.P. Also, find thecommon ratio in each case:

(i) 4, -2, 1, -1/2, ….

(ii) -2/3, -6, -54, ….

(iii) a, 3a2/4, 9a3/16, ….

(iv) ½, 1/3, 2/9, 4/27, …



Answer -

(i) 4, -2, 1, -1/2, ….

Let a = 4, b = -2, c =1

In GP,

b2 =ac

(-2)2 =4(1)

4 = 4

So, the Common ratio =r = -2/4 = -1/2

(ii) -2/3, -6, -54, ….

Let a = -2/3, b = -6,c = -54

In GP,

b2 =ac

(-6)2 =-2/3 × (-54)

36 = 36

So, the Common ratio =r = -6/(-2/3) = -6 × 3/-2 = 9

(iii) a, 3a2/4,9a3/16, ….

Let a = a, b = 3a2/4,c = 9a3/16

In GP,

b2 =ac

(3a2/4)2 =9a3/16 × a

9a4/4 = 9a4/16

So, the Common ratio =r = (3a2/4)/a = 3a2/4a = 3a/4

(iv) ½, 1/3, 2/9, 4/27, …

Let a = 1/2, b = 1/3,c = 2/9

In GP,

b2 =ac

(1/3)2 =1/2 × (2/9)

1/9 = 1/9

So, the Common ratio =r = (1/3)/(1/2) = (1/3) × 2 = 2/3

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