The sum of first 20 terms of the sequence 0.7 ,0.77 , 0.777 …., is

Options

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

Answer

C

Explanation

Let `S=0.7+0.77+0.777+ . . . +` upto 20 terms
Then,
`S=(7)/(10)+(77)/(100)+(777)/(1000)+. . . . ` upto 20 terms
`rArr" "S=7{(1)/(10)+(11)/(100)+(111)/(1000)+ . . . ." upto 20 terms"}`
`rArr" "S=(7)/(9){(9)/(10)+(99)/(100)+(999)/(1000)+ . . . ." upto 20 terms"}`
`rArr" "S=(7)/(9){(1-(1)/(10))+(1-(1)/(100))+(1-(1)/(1000))+ . . . +(1-(1)/(10^(20)))}`
`rArr" "S=(7)/(9){20-((1)/(10)+(1)/(10^(2))+ . . . .+(1)/(10^(20)))}`
`rArr" "S=(7)/(9){20-(1)/(10)((1-(1)/(10^(20)))/(1-(1)/(10)))}`
`rArr" "S=(7)/(9){20-(1)/(9)(1-(1)/(10^(20)))}`
`rArr" "S=(7)/(9){(179)/(9)+(1)/(9)((1)/(10))^(20)}=(7)/(81)(179+10^(-20))`