Which equation could be used to calculate the sum of the geometric series? (1)/(3)+(2)/(9)+(4)/(27)+(8)/(81)+(16)/(243) S_(5)-(frac (1)/(3)(1-((2)/(3))^5))((1-(2)/(3))) S_(5)=(frac (2)/(3)(1-((1)/(3))^5))((1-(1)/(3)))

To calculate the sum of the given geometric series, we can use the formula for the sum of a finite geometric series:

$$S_n = \frac{a(1-r^n)}{1-r}$$

where $$S_n$$ is the sum of the first $$n$$ terms, $$a$$ is the first term, $$r$$ is the common ratio, and $$n$$ is the number of terms.

In this case, the first term $$a$$ is $$\frac{1}{3}$$ , the common ratio $$r$$ is $$\frac{2}{3}$$ , and the number of terms $$n$$ is 5.

Substituting these values into the formula, we get:

$$S_5 = \frac{\frac{1}{3}(1-(\frac{2}{3})^5)}{1-\frac{2}{3}}$$

Therefore, the correct equation to calculate the sum of the given geometric series is:

$$S_5 = \frac{\frac{1}{3}(1-(\frac{2}{3})^5)}{1-\frac{2}{3}}$$