Q. 85.0( 2 Votes )

# Find the sum of first n terms of the series

(i) 7 + 77 + 777 + … . (ii) 0.4 + 0.94 + 0.994 + … .

Answer :

(i) Series = 7 + 77 + 777 + …..n 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 + ….. – (n×1)]

⇒ Series = [10 + 100 + 1000 + …..] –

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

First term = 10

Common ratio = = 10

Sum of n terms =

⇒ Sum of n terms =

⇒ Sum of n terms =

⇒ Series = [] –

⇒ Series = –

(ii) Series = 0.4 + 0.94 + 0.994 + …..n terms

Series = (1–0.6) + (1–0.06) + (1–0.006) + ……..

⇒ Series = n×1 – (0.6 + 0.06 + 0.006 + ….)

We find the sum of 0.6 + 0.06 + 0.006…..n terms as:

First term = 0.6

Common ratio = =

Sum of n terms =

⇒ Sum of n terms =

⇒ Sum of n terms =

⇒ Series = n –

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