Question

Question 21 Find the sum of the following series up to n terms:

(i) 5 + 55 + 555 + …

(ii) .6 +.66 +. 666 +…

Class X1 - Maths -Sequences and Series Page 200

Answer 1

(1) Let S = 5 + 55 + 555 + ......... n terms
= 5[ 1 + 11 + 111 + ............ n terms ]
= 5/9[ 9 + 99 + 999 + ........... n terms ]
= 5/9 [ (10 - 1) + (100 - 1) + (1000 - 1) + .... n terms]
=5/9[ (10 + 10² + 10³ + ... n terms ) - (1 + 1 + .....n terms ) ]

use the formula,
Sₙ = a( rⁿ - 1 )/( r - 1) , r > 1

= 5/9 [ 10(10ⁿ - 1)/(10 - 1) - n ]
= 5/9 [ 10(10ⁿ - 1)/9 - n ]


(2) Let S = 0.6 + 0.66 + 0.666 + ......+ n terms
= 6[0.1 + 0.11 + 0.111 + ....... + n terms ]
= 6/9[0.9 + 0.99 + 0.999 + ......... + n terms ]
= 2/3 [ (1 - 0.1) + (1 - 0.01) + (1 - 0.001) + .....+ n terms ]
= 2/3 [ (1 + 1 + 1 ... + n terms ) - ( 0.1 + 0.01 + 0.001 + .... n terms )]

use the formula,
Sₙ = a( 1 - rⁿ )/( 1 - r ) , r < 1

= 2/3 [ n - 0.1{ 1 - (0.1)ⁿ}/(1 - 0.1)]
= 2/3 [ n - 1/9 { 1 - (1/10)ⁿ } ]