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]
= \(\frac{8}{9}\)(9 + 99+ 999 + … to n terms)
=\(\frac{8}{9}\)((10-1) + (100-1) + (1000-1) + … to n terms )
=\(\frac{8}{9}\) ( ( 10 + 100 + 1000 + … to n terms) – ( 1+1+1+ … to n terms)
= \(\frac{8}{9}\)( ( 10 + 100 + 1000 + … to n terms) – n)
Sum of a G.P. series is represented by the formula, Sn = \(a \frac{r^n - 1}{r-1}\), when r>1.
‘Sn’ represents the sum of the G.P. series up to 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
= \(\frac{8}{9}\)( ( 10 + 100 + 1000 + … to n terms) – n)
= \(\frac{8}{9} \big( \frac{10^{n+1}-10}{9} - n \big)\)
(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 ]
= \(\frac{3}{9}\)( 9+99+999+ … to n terms )
= \(\frac{3}{9}\)( (10-1) + (100-1) + (1000-1) + … to n terms )
= \(\frac{3}{9}\)( ( 10+100+1000+ …to n terms ) – (1+1+1+ … to n terms) )
= \(\frac{3}{9}\)( (10+100+1000+ to n terms) – n )
Sum of a G.P. series is represented by the formula, Sn = \(\frac{r^n - 1}{r-1}\) , when r>1.
‘Sn’ represents the sum of the G.P. series up to 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
= \(\frac{3}{9}\)( (10+100+1000+ to n terms) – n )
= \(\frac{3}{9}\) \(\big(\frac{10^{n+1} - 10}{9} - n\big)\)
(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 ]
= \(\frac{7}{9}\)( 0.9+0.99+0.999+ … to n terms )
= \(\frac{7}{9}\)( (1-0.1)+(1-0.01)+(1-0.001)+ … to n terms)
= \(\frac{7}{9}\)( (1+1+1+ … to n terms )–(0.1+0.01+0.001+… to n terms ))
= \(\frac{7}{9}\)( n – (0.1+0.01+0.001+ … to n terms ) )
Sum of a G.P. series is represented by the formula, Sn = a\(\frac{1-r^n}{1-r}\), when |r|<1.
‘Sn’ represents the sum of the G.P. series up to 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]
⇒ Sn = \(\frac{1-0.1^n}{9}\)
∴ The sum of the given expression is
= \(\frac{7}{9}\)( n – (0.1+0.01+0.001+ … to n terms ) )
= \(\frac{7}{9}\)( n – (\(\frac{1-0.1^n}{9}\)) )