To find which equation could be used to calculate the sum of the given geometric series, let's follow these steps:
1. Identify the first term ([tex]\(a\)[/tex]) and the common ratio ([tex]\(r\)[/tex]) of the geometric series:
[tex]\[
\frac{1}{3}, \frac{2}{9}, \frac{4}{27}, \frac{8}{81}, \frac{16}{243}
\][/tex]
The first term [tex]\(a\)[/tex] is the first term of the series:
[tex]\[
a = \frac{1}{3}
\][/tex]
To find the common ratio [tex]\(r\)[/tex], we divide the second term by the first term:
[tex]\[
r = \frac{\frac{2}{9}}{\frac{1}{3}} = \frac{2}{9} \times \frac{3}{1} = \frac{2}{3}
\][/tex]
2. Verify that the common ratio is consistent:
Let's quickly verify the common ratio [tex]\(r = \frac{2}{3}\)[/tex] with the next terms:
[tex]\[
\frac{4}{27} \div \frac{2}{9} = \frac{4}{27} \times \frac{9}{2} = \frac{4 \times 9}{27 \times 2} = \frac{36}{54} = \frac{2}{3}
\][/tex]
[tex]\[
\frac{8}{81} \div \frac{4}{27} = \frac{8}{81} \times \frac{27}{4} = \frac{8 \times 27}{81 \times 4} = \frac{216}{324} = \frac{2}{3}
\][/tex]
[tex]\[
\frac{16}{243} \div \frac{8}{81} = \frac{16}{243} \times \frac{81}{8} = \frac{16 \times 81}{243 \times 8} = \frac{1296}{1944} = \frac{2}{3}
\][/tex]
Since the common ratio [tex]\( r = \frac{2}{3} \)[/tex] is consistent, we can use the sum formula for a geometric series.
3. Use the sum formula for the first [tex]\(n\)[/tex] terms of a geometric series:
The sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of a geometric series is given by:
[tex]\[
S_n = a \frac{1 - r^n}{1 - r}
\][/tex]
Given [tex]\( n = 5 \)[/tex], [tex]\( a = \frac{1}{3} \)[/tex], and [tex]\( r = \frac{2}{3} \)[/tex], we substitute these values into the formula:
[tex]\[
S_5 = \frac{\frac{1}{3} \left( 1 - \left( \frac{2}{3} \right)^5 \right)}{1 - \frac{2}{3}}
\][/tex]
4. Verify the other equation:
Comparing this with your given options:
[tex]\[
S_5 = \frac{\frac{1}{3} \left( 1 - \left( \frac{2}{3} \right)^5 \right)}{1 - \frac{2}{3}}
\][/tex]
The other equation provided is:
[tex]\[
S_5 = \frac{\frac{2}{3} \left( 1 - \left( \frac{1}{3} \right)^5 \right)}{1 - \frac{1}{3}}
\][/tex]
Clearly, the correct form matches the first option, not the second one.
Thus, the equation that could be used to calculate the sum of the given geometric series is:
[tex]\[
S_5 = \frac{\frac{1}{3} \left(1 - \left(\frac{2}{3}\right)^5 \right)}{1 - \frac{2}{3}}
\][/tex]
