Sum of the series `0.5+ 0.55+ 0.555 +... ` upto n terms is

Options

A. `(5)/(9)[n-(2)/(9)(1-(1)/(10^(n)))]`
B. `(1)/(9)[5-(2)/(9)(1-(1)/(10^(n)))]`
C. `(1)/(9)[n-(5)/(9)(1-(1)/(10^(n)))]`
D. `(5)/(9)[n-(1)/(9)(1-(1)/(10^(n)))]`

Answer

D

Explanation

Given `0.5+0.55+0.555+......`to n
`=5[0.1+0.11+0.111+......"to n terms"]`
`(5)/(9)[0.9+0.99+0.999+......"to n terms"]`
`=(5)/(9)[(9)/(10)+(99)/(100)+(999)/(1000)+......"to n terms"]`
`(5)/(9)[(1-(1)/(10))+(1-(1)/(100))+(1-(1)/(1000))+......"to n terms"]`
`=(5)/(9)[(1-(1)/(10))+(1-(1)/(10^(2)))+(1-(1)/(10^(3)))+......+(1-(1)/(10^(n)))]`
`=(5)/(9)[n-((1)/(10)+(1)/(10^(2))+...(1)/(10^(n)))]`
`=(5)/(9)[n-(1)/(10)({1-((1)/(10))^(n)})/((1-(1)/(10)))]`
`=(5)/(9)[n-(1)/(9)(1-(1)/(10^(n)))]`