Let's analyze the given 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] of the series is [tex]\( \frac{1}{3} \)[/tex]. The common ratio [tex]\( r \)[/tex] is obtained by dividing the second term by the first term, i.e.,
[tex]\[
r = \frac{\frac{2}{9}}{\frac{1}{3}} = \frac{2}{9} \times \frac{3}{1} = \frac{2}{3}
\][/tex]
We are asked to find the sum of the first 5 terms of this geometric series. The sum of the first [tex]\( n \)[/tex] terms of a geometric series is given by the formula:
[tex]\[
S_n = \frac{a (1 - r^n)}{1 - r}
\][/tex]
Substituting the values [tex]\( a = \frac{1}{3} \)[/tex], [tex]\( r = \frac{2}{3} \)[/tex], and [tex]\( n = 5 \)[/tex] into the formula, we get:
[tex]\[
S_5 = \frac{\frac{1}{3} (1 - (\frac{2}{3})^5)}{1 - \frac{2}{3}}
\][/tex]
Now let's simplify step-by-step:
1. Calculate [tex]\( (\frac{2}{3})^5 \)[/tex]:
[tex]\[
(\frac{2}{3})^5 = \frac{2^5}{3^5} = \frac{32}{243}
\][/tex]
2. Subtract this value from 1:
[tex]\[
1 - \frac{32}{243} = \frac{243}{243} - \frac{32}{243} = \frac{211}{243}
\][/tex]
3. Multiply this by [tex]\( \frac{1}{3} \)[/tex]:
[tex]\[
\frac{1}{3} \cdot \frac{211}{243} = \frac{211}{729}
\][/tex]
4. Simplify the denominator [tex]\( 1 - \frac{2}{3} \)[/tex]:
[tex]\[
1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}
\][/tex]
5. Divide the result from step 3 by the result from step 4:
[tex]\[
S_5 = \frac{\frac{211}{729}}{\frac{1}{3}} = \frac{211}{729} \times 3 = \frac{211 \times 3}{729} = \frac{633}{729}
\][/tex]
To simplify [tex]\( \frac{633}{729} \)[/tex]:
[tex]\[
\frac{633 \div 3}{729 \div 3} = \frac{211}{243}
\][/tex]
However, in numerical form, the result is approximately:
[tex]\[
S_5 \approx 0.8683127572016459
\][/tex]
Thus, the sum of the first 5 terms of the given geometric series is approximately:
[tex]\[
S_5 \approx 0.8683
\][/tex]
