The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, .. , is (1) `7/9(99-10^(-20))` (2) `7/(81)(179+10^(-20))` (3) `7/9(99+10^(-20))` (3) `7/(81)(179-10^(-20))`

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

0.7+0.77+0.777+….+0.7777….7
`=7/9[0.9+0.99+0.999+….+0.999…9]`
`=7/9[(1-0.1)+(1-0.01)+(1-0.001)+….+(1-0.000…1)]`
`=7/9[20-1/10cdot(1-1/10^(20))/(1-1/10)]=7/9[20-1/9cdot((10^(20)-1)/10^(20))]`
`=7/81[180-(1-1/10^(20))]=7/81[179+10^(-20)]`