Find the sum of the following series up to n terms: (i) `5" "+" "55" "+" "555" "+" "dot" "dot" "dot` (ii) `. 6" "+dot" "66" "+dot" "666" "+dot" "dot" "dot`

Explanation

We have,
`5+55+555+`.. to n terms
`=5[1+11+111+`…. To n terms]
`=5/9[9+99+999+`.. To n terms]
`=5/9[(10-1)+(10^(2)-1)+(10^(3)-1)+…+(10^(n)-1)]`
`=5/9[10xx((10n-1))/(10-1)-n]`
`=5/9[10/9(10n-1)-n]`
`=5/81[10^(n+1)-10-9n]`
(ii) Let
`S_(n)=0.6+0.66+0.666+...` to n terms
`=6[0.1+0.11+0.111+ ...` to n terms]
`=6/9[0.9+0.99+0.999+`.... to n terms]
`=6/9[(1-1/10)+(1-1/(10^(2)))+(1-1/(10^(3)))+...` n terms]
`=2/3[(1+1+...`n terms)-`1/10(1+1/10+1/(10^(2))+..`n terms)]
`=2/3[n-1/10((1-(1/10)^(n))/(1-1/10))]`
`=2/3n-2/3xx1/9(1-10^(-n))`
`=2/3n-2/27(1-10^(-n))`