Find the sum of the series :
NOTE: The following terms are not G.P. series, but we can convert them to form one.
(i) 8 + 88 + 888 + …. To n terms
(ii) 3 + 33 + 333 + …. To n terms
(iii) 0.7 + 0.77 + 0.777 + …. To n terms
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. ‘Sn’ represents the sum of the G.P. series upto nth 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. ‘Sn’ represents the sum of the G.P. series upto nth 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. ‘Sn’ represents the sum of the G.P. series upto nth 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 – ( 
) )