0.2 + 0.22 + 0.222 + ..... to n terms
= 2[0.1 + 0.11 + 0.111 + ..... to n terms]
= \(\frac{2}{9}\) [0.9 + 0.99 + 0.999 + ..... to n terms]
= \(\frac{2}{9}\) [(1 – 0.1) + (1 – 0.01) + (1 + 0.001) + ..... to n terms]
= \(\frac{2}{9}\) [(1 + 1 + 1 + ..... to n terms) – (0.1 + 0.01 + 0.001 + ..... to n terms)]
= \(\frac{2}{9}\)\(\bigg[n-\frac{0.1(1-(0.1)^n}{(1-0.1)}\bigg]\) = \(\frac{2}{9}\)\(\bigg[n-\frac{\frac{1}{10}\big(1-\frac{1}{10^n}\big)}{1-\frac{1}{10}}\bigg]\) = \(\frac{2}{9}\)\(\bigg[n-\frac{1}{9}\bigg(1-\frac{1}{10^n}\bigg)\bigg]\).