To calculate the sum of the given geometric series:
[tex]\[
\frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \frac{8}{81} + \frac{16}{243}
\][/tex]
we can follow a step-by-step process to identify the common characteristics of the series and find the appropriate formula to calculate its sum.
### Step-by-Step Solution:
1. Identify the first term ([tex]\(a\)[/tex]):
- The first term of the series is [tex]\(\frac{1}{3}\)[/tex].
2. Determine the common ratio ([tex]\(r\)[/tex]):
- The common ratio [tex]\(r\)[/tex] is found by dividing the second term by the first term:
[tex]\[
r = \frac{\frac{2}{9}}{\frac{1}{3}} = \frac{2/9}{1/3} = \frac{2/9 \cdot 3}{1} = \frac{2}{3}
\][/tex]
- Verifying for the subsequent terms:
[tex]\[
r = \frac{\frac{4}{27}}{\frac{2}{9}} = \frac{4/27}{2/9} = \frac{4/27 \cdot 9}{2} = \frac{4}{6} = \frac{2}{3}
\][/tex]
[tex]\[
r = \frac{\frac{8}{81}}{\frac{4}{27}} = \frac{8/81}{4/27} = \frac{8/81 \cdot 27}{4} = \frac{8}{12} = \frac{2}{3}
\][/tex]
3. Identify the number of terms ([tex]\(n\)[/tex]):
- The series has 5 terms.
4. Sum of 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]\[
S_n = a \left(\frac{1 - r^n}{1 - r}\right)
\][/tex]
- Substitute the values [tex]\(a = \frac{1}{3}, r = \frac{2}{3}, n = 5\)[/tex]:
[tex]\[
S_5 = \frac{1}{3} \left(\frac{1 - \left(\frac{2}{3}\right)^5}{1 - \frac{2}{3}}\right)
\][/tex]
5. Calculate the individual components:
- Calculate [tex]\(r^5\)[/tex]:
[tex]\[
r^5 = \left(\frac{2}{3}\right)^5 = \frac{32}{243}
\][/tex]
- Subtract [tex]\(r^5\)[/tex] from 1:
[tex]\[
1 - r^5 = 1 - \frac{32}{243} = \frac{243}{243} - \frac{32}{243} = \frac{211}{243}
\][/tex]
- Calculate the denominator [tex]\(1 - r\)[/tex]:
[tex]\[
1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}
\][/tex]
- Compute the sum:
[tex]\[
S_5 = \frac{1}{3} \left(\frac{\frac{211}{243}}{\frac{1}{3}}\right) = \frac{1}{3} \times \frac{211}{243} \times 3 = \frac{211}{243} = 0.8683127572016459
\][/tex]
Therefore, the equation used to calculate the sum of the geometric series is:
[tex]\[
S_n = a \left(\frac{1 - r^n}{1 - r}\right)
\][/tex]
