(c) \(\frac{7}{81}(179+10^{-20})\)
Let S20 = 0.7 + 0.77 + 0.777 + ..... upto 20 terms
= 7(0.1 + 0.11 + 0.111 + ..... upto 20 terms)
= \(\frac{7}{9}\) (0.9 + 0.99 + 0.999 + ..... upto 20 terms)
= \(\frac{7}{9}\) [(1 – 0.1) + (1 – 0.01) + (1 – 0.001) + upto 20 terms)
= \(\frac{7}{9}\) [(1 + 1 + 1 + ..... upto 20 terms) – (0.1 + 0.01 + 0.001 + ..... upto 20 terms)]
= \(\frac{7}{9}\)\(\bigg[20-\frac{0.1\{1-(0.1)^2\}}{(0-0.1)}\bigg]\) \(\bigg(\because\,S_n=\frac{a(1-r^n)}{1-r}, \text{when}\,r<1\bigg)\)
= \(\frac{7}{9}\)\(\bigg[20-\frac{1}{9}\bigg(1-\big({\frac{1}{10}\big)^{20}}\bigg)\bigg]\) = \(\frac{7}{9}\)\(\bigg[20-\frac{1}{9}+\frac{10}{9}^{-20}\bigg]\)
= \(\frac{7}{9}\)\(\bigg[\frac{179+10^{-20}}{9}\bigg]\) = \(\frac{7}{81}(179+10^{-20})\).