Question

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))`

Answer 1

`s= 0.7 + 0.77 + 0.777 + .... 0.7777 ` 20 times
`= 7(0.1 + 0.11 + 0.11 + 0.111 + ....... + 0.1111 ....)`
= `7/9[0.9 + 0.99 + 0.99 + 0.999 + ...........)`
`= 7/9[(1-0.1) + (1-0.01) + ....+ (1-0.000......1)]`
`=7/9 [ 20 - (1/10 + 1/10^2 + ......1/20^20)]`
here, `a=1/10 , r= 1/10`
`=7/9[ 20 - 1/10(1-1/10^20)/(1- 1/10)`
`= 7/9 [20 -1/9 (1-10^-20)`
`=7/81[179 + 10^-20]`
Answer