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`

Answer

N/a

Explanation

(i) 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) + …` to 'n' terms
`=(5)/(9) [(10 + 10^(2) + 10^(3) + …
" to " 'n' " terms ") - n]`
`=(5)/(9) [(10(10^(n) - 1))/(10 - 1) - n] =(5)/(9) [(10(10^(n) - 1))/(9) - n]*`
(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 - 0.1) + (1 - 0.01) + (1 - 0.001) + ... + " to n terms "]`
`= (2)/(3) [n - {(0.1) + (0.1)^(2) + (0.1)^(3) + ... + (0.1)^(n)}]`
`= (2)/(3)n-(2)/(3) [(0.1) + (0.1)^(2) + (0.1)^(3) + ... + (0.1)^(n)]`
`= (2)/(3)n-(2)/(3) [(0.1{1 -(0.1)^(n)})/(1 - 0.1)]`
`= (2)/(3)n-(2)/(27) [1 -(0.1)^(n)] = (2)/(3)n-(2)/(27) [1 -(10)^(-n)]`
Therefore, `S_(n) = (2)/(3)n-(2)/(27) [1 -(10)^(-n)]`