Let's write out the formula for the sum of geometric series,
[tex]S_n=\frac{a(1-r^n)}{1-r}[/tex][tex]\begin{gathered} \text{where a=first term=729} \\ r=\text{common ratio=}\frac{1}{-3} \\ S_7=?,\text{ where n= number of terms=7} \end{gathered}[/tex]Let's solve for the sum of seven geometric series,
[tex]\begin{gathered} S_7=\frac{729(1-(-\frac{1}{3})^7)}{1-(-\frac{1}{3})} \\ =\frac{729(1-(-\frac{1}{2187}))}{1+\frac{1}{3}} \\ =\frac{729(1+\frac{1}{2187})}{\frac{4}{3}} \end{gathered}[/tex][tex]\begin{gathered} S_7=\frac{729(1\frac{1}{2187})}{\frac{4}{3}} \\ =\frac{729(\frac{2188}{2187})}{\frac{4}{3}}=\frac{\frac{2188}{3}}{\frac{4}{3}} \\ =\frac{2188}{3}\frac{\text{.}}{.}\frac{4}{3} \\ =\frac{2188}{3}\times\frac{3}{4}=547 \end{gathered}[/tex]Hence,
[tex]S_7=547[/tex]Therefore, option 5 is the correct answer.
Option 5 is none of these are correct.
