The expression can be rewritten as

[ Taking 8 as a common factor ]

8( 1+ 11 + 111+ … to n terms)

[Multiplying and dividing the expression by 9]

= ( 9 + 99+ 999 + … to n terms)

= ( (10-1) + (100-1) + (1000-1) + … to n terms )

= ( ( 10 + 100 + 1000 + … to n terms) – ( 1+1+1+ … to n terms)

= ( ( 10 + 100 + 1000 + … to n terms) – n)

Sum of a G.P. series is represented by the formula, , when r>1. ‘S_n’ represents the sum of the G.P. series upto n^th terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.

Here,

a = 10

r = (ratio between the n term and n-1 term) 10

n terms

⇒

⇒

∴ The sum of the given expression is

= ( ( 10 + 100 + 1000 + … to n terms) – n)

= ()

(ii) The given expression can be rewritten as

[ taking 3 common ]

= 3( 1+11+111+ …to n terms)

[ multiplying and dividing the expression by 9 ]

= ( 9+99+999+ … to n terms )

= ( (10-1) + (100-1) + (1000-1) + … to n terms )

= ( ( 10+100+1000+ …to n terms ) – (1+1+1+ … to n terms) )

= ( (10+100+1000+ to n terms) – n )

Sum of a G.P. series is represented by the formula, , when r>1. ‘S_n’ represents the sum of the G.P. series upto n^th terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.

Here,

a = 10

r = (ratio between the n term and n-1 term) 10

n terms

⇒

⇒

∴ The sum of the given expression is

= ( (10+100+1000+ to n terms) – n )

= ( - n )

(iii) We can rewrite the expression as

[ taking 7 as a common factor]

= 7(0.1+0.11+0.111+ … to n terms)

[ multiplying and dividing by 9 ]

= ( 0.9+0.99+0.999+ … to n terms )

= ( (1-0.1)+(1-0.01)+(1-0.001)+ … to n terms)

= ( (1+1+1+ … to n terms )–(0.1+0.01+0.001+… to n terms ))

= ( n – (0.1+0.01+0.001+ … to n terms ) )

Sum of a G.P. series is represented by the formula, , when |r|<1. ‘S_n’ represents the sum of the G.P. series upto n^th terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.

Here,

a = 0.1

r = (ratio between the n term and n-1 term) 0.1

n terms

⇒

[multiplying both numerator and denominator by 10]

⇒

∴ The sum of the given expression is

= ( n – (0.1+0.01+0.001+ … to n terms ) )

= ( n – ( ) )