(i) Let $S_n$ = 5 + 55 + 555 + ..... to n terms, which can be written as $S_n$ = $\frac{5}{9}$ [9 + 99 + 999 + ... to n terms] = $\frac{5}{9}$ [(10 - 1) + (100 - 1) + (1000 - 1) + ... to n terms] = $\frac{5}{9}$ [(10 + 100 + 1000 + ... to n terms) - (1 + 1 + ... to n terms)] = $\frac{5}{9}\left[\frac{10(10^n-1)}{10-1} - n\right]$ = $\frac{5}{9}\left[\frac{10}{9}(10^n-1) - n\right]$ = $\frac{50}{81}(10^n-1) - \frac{5n}{9}$ (ii) Let $S_n$ = 0.6 + 0.66 + 0.666 + ...... to n terms, which can be written as $S_n$ = $\frac{6}{9}$ [0.9 + 0.99 + 0.999 + ... to n terms] = $\frac{6}{9}$ [(1 - 0.1) + (1 - 0.01) + (1 - 0.001) + ... to n terms] = $\frac{6}{9}$ [(1 + 1 + ... to n terms) - (0.1 + 0.01 + 0.001 + ... to n terms)] = $\frac{2}{3}\left[n - 0.1\frac{1-(0.1)^n}{1-0.1}\right]$ = $\frac{2}{3}\left[n - \frac{1}{9}\left(1-\frac{1}{10^n}\right)\right]$ = $\frac{2n}{3} - \frac{2}{27}\left(1-\frac{1}{10^n}\right)$