(b) 2059
Let the first term and common ratio of the G.P. be a and r respectively.
Then, T1 = a = 729, T7 = ar6 = 64
∴ \(\frac{ar^6}{a}\) = \(\frac{64}{729}\) ⇒ r6 = \(\frac{64}{729}\) = \(\frac{2^6}{3^6}\) ⇒ r = \(\frac{2}{3}.\)
∴ S7 = \(\frac{a(1-r^n)}{1-r}\) = \(\frac{729\bigg(1-\big(\frac{2}{3}\big)^7\bigg)}{1-\frac{2}{3}}\)
= \(\frac{729\bigg(1-\frac{128}{2187}\bigg)}{\frac{1}{3}}\) = 2187 \(\bigg(\frac{2187-128}{2187}\bigg)\) = 2059.