To find the sum of the first 6 terms of the given geometric series [tex]\(6 + (-18) + 54 + (-162) + \ldots\)[/tex], we will use the formula for the sum of the first [tex]\(n\)[/tex] terms of a geometric series:
[tex]\[
S_n = \frac{a_1 \left(1 - r^n\right)}{1 - r}
\][/tex]
where:
- [tex]\(a_1\)[/tex] is the first term of the series
- [tex]\(r\)[/tex] is the common ratio
- [tex]\(n\)[/tex] is the number of terms
Let's identify the values from the series:
1. The first term, [tex]\(a_1\)[/tex], is [tex]\(6\)[/tex].
2. The common ratio, [tex]\(r\)[/tex], can be found by dividing the second term by the first term:
[tex]\[ r = \frac{-18}{6} = -3 \][/tex]
3. The number of terms, [tex]\(n\)[/tex], is [tex]\(6\)[/tex].
Plug these values into the formula to compute the sum:
[tex]\[
S_6 = \frac{6 \left(1 - (-3)^6\right)}{1 - (-3)}
\][/tex]
Calculate [tex]\((-3)^6\)[/tex]:
[tex]\[
(-3)^6 = (-3) \times (-3) \times (-3) \times (-3) \times (-3) \times (-3) = 729
\][/tex]
Substitute this back into the formula:
[tex]\[
S_6 = \frac{6 \left(1 - 729\right)}{1 + 3}
\][/tex]
Simplify inside the parentheses:
[tex]\[
1 - 729 = -728
\][/tex]
Then the formula becomes:
[tex]\[
S_6 = \frac{6 \times (-728)}{4}
\][/tex]
Multiply and divide:
[tex]\[
S_6 = \frac{-4368}{4} = -1092
\][/tex]
Thus, the sum of the first 6 terms of the geometric series is [tex]\(-1092\)[/tex].
Therefore, the correct answer is:
A. [tex]\(-1,092\)[/tex]
