The Series \(\frac{2}{3}\) + \(\frac{8}{9}\) + \(\frac{26}{27}\) + \(\frac{80}{81}\).....to n terms can be written as
\(\big(1-\frac{1}{3}\big)\) + \(\big(1-\frac{1}{9}\big)\) + \(\big(1-\frac{1}{27}\big)\) + \(\big(1-\frac{1}{81}\big)\) + ...... to n terms
= (1 + 1 + 1 + 1 + .... to n terms)- \(\bigg(\frac13+\frac19+\frac{1}{27}+\frac{1}{81}+....\text{to n terms}\bigg)\)
= n - \(\frac{\frac13\bigg(1-\big(\frac{1}{3}\big)^n\bigg)}{1-\frac13}\) \(\big(S_n = \frac{a(1-r^n)}{1-r}, \text{here}\,a=\frac{1}{3},\,r=\frac{1}{3}\big)\)
= n - \(\frac{\frac13\bigg(1-\frac{1}{3^n}\bigg)}{\frac23}\) = n - \(\frac{1}{2}(1-3^{-n}).\)
