The sum of 10 terms of the series `0.7+0.77+0.777+........` is

Options

A. `(7)/(81) (179 - 10^(-20))`
B. `(7)/(9) (99 -10^(-20))`
C. `(7)/(81) (179 + 10^(-20))`
D. `(7)/(9) (99 + 10^(-20))`

Answer

C

Explanation

Let `S = 0.7 + 0.77 + 0.777 +`...
`= (7)/(10) + (77)/(10^(2)) + (777)/(10^(3)) +`... upto 20 terms
`= 7 [(1)/(10) + (11)/(10^(2)) + (111)/(10^(3))+ " ... upto 20 terms"]`
`= (7)/(9) [(9)/(10) + (99)/(100) + (999)/(1000) + "... upto 20 terms"]`
`= (7)/(9) [(1 - (1)/(10)) + (1 -(1)/(10^(2))) + (1- (1)/(10^(3))) + ..+ "upto 20 terms"]` `= (7)/(9) [(1 + 1 +...+ " upto 20 terms") - ((1)/(10) + (1)/(10^(2)) + (1)/(10^(3)) + ...+ " upto 20 terms")]`
`= (7)/(9) [20 - ((1)/(10) {1 - ((1)/(10))^(20)})/(1 - (1)/(10))]`
`[( :' underset(i=1)overset(20)(Sigma)=20" and sum of n terms of"),("GP,"S_(n)=(alpha(1-r^(n)))/(1-r)" when "(r lt 1))]`
`= (7)/(9) [20 - (1)/(9) {1 - ((1)/(10))^(20)}]`
`= (7)/(9) [(179)/(9) + (1)/(9) ((1)/(10))^(20)] = (7)/(81) [179 + (10)^(-20)]`