Let's determine the sum of the first five terms of the given geometric series:
[tex]\[3 + (-9) + 27 + (-81) + \cdots\][/tex]
First, identify the first term [tex]\((a)\)[/tex] and the common ratio [tex]\((r)\)[/tex] of the geometric series:
- The first term [tex]\(a\)[/tex] is 3.
- The common ratio [tex]\(r\)[/tex] can be found by dividing the second term by the first term:
[tex]\[
r = \frac{-9}{3} = -3
\][/tex]
We use the formula for the sum of the first [tex]\(n\)[/tex] terms of a geometric series:
[tex]\[
S_n = a \left( \frac{1 - r^n}{1 - r} \right) \quad \text{for} \quad r \neq 1.
\][/tex]
For this problem, we want the sum of the first five terms [tex]\((n = 5)\)[/tex]:
[tex]\[
a = 3, \quad r = -3, \quad n = 5
\][/tex]
Plug these values into the formula:
[tex]\[
S_5 = 3 \left( \frac{1 - (-3)^5}{1 - (-3)} \right)
\][/tex]
Calculate [tex]\((-3)^5\)[/tex]:
[tex]\[
(-3)^5 = -243
\][/tex]
Now substitute this back into the formula:
[tex]\[
S_5 = 3 \left( \frac{1 - (-243)}{1 + 3} \right)
\][/tex]
[tex]\[
S_5 = 3 \left( \frac{1 + 243}{4} \right)
\][/tex]
[tex]\[
S_5 = 3 \left( \frac{244}{4} \right)
\][/tex]
[tex]\[
S_5 = 3 \times 61
\][/tex]
[tex]\[
S_5 = 183
\][/tex]
Therefore, the sum of the first five terms is:
[tex]\[
\boxed{183}
\][/tex]
