Find the sum of the following finite series

(i) 1 + 0.1 + 0.01 + 0.001 + … + ( 0.1)9 (ii) 1 + 11 + 111 + … to 20 terms.

(i) In the G.P.,

First term = 1

Common ratio = = 0.1

Sum of n terms =

⇒ Sum of 10 terms =

⇒ Sum of 10 terms =

(ii) Series = 1 + 11 + 111 + …..20 terms

Series = [9×(1 + 11 + 111 + …..)] ( Multiplying and dividing by 9)

⇒ Series = [9 + 99 + 999 + …..]

⇒ Series = [(10–1) + (100–1) + (1000–1) + …..]

⇒ Series = [10 + 100 + 1000 + ….. – (20×1)]

⇒ Series = [10 + 100 + 1000 + …..] –

We find the sum of 10 + 100 + 1000…..20terms as:

First term = 10

Common ratio = = 10

Sum of n terms =

⇒ Sum of 20 terms =

⇒ Sum of 20 terms =

⇒ Series = [] –

⇒ Series = –