To evaluate the geometric series described by the terms [tex]\(-4, -16, -64, -256, \ldots\)[/tex] with [tex]\(n=6\)[/tex], let's break down the steps for calculating the sum of the series:
1. Identify the terms of the geometric series:
- The first term ([tex]\(a\)[/tex]) is [tex]\(-4\)[/tex].
- The common ratio ([tex]\(r\)[/tex]) can be found by dividing the second term by the first term:
[tex]\[
r = \frac{-16}{-4} = 4
\][/tex]
2. Set up the formula for the sum 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]\[
S_n = a \frac{1-r^n}{1-r}, \text{ where } r \neq 1
\][/tex]
In this case, [tex]\(a = -4\)[/tex], [tex]\(r = 4\)[/tex], and [tex]\(n = 6\)[/tex].
3. Substitute the values into the formula and compute:
[tex]\[
S_6 = -4 \frac{1-4^6}{1-4}
\][/tex]
Calculate [tex]\(4^6\)[/tex]:
[tex]\[
4^6 = 4096
\][/tex]
Then substitute this into the formula:
[tex]\[
S_6 = -4 \frac{1-4096}{1-4}
\][/tex]
Simplify the fraction:
[tex]\[
1-4096 = -4095
\][/tex]
[tex]\[
1-4 = -3
\][/tex]
So,
[tex]\[
S_6 = -4 \frac{-4095}{-3} = -4 \times 1365
\][/tex]
Finally, calculate the product:
[tex]\[
S_6 = -5460
\][/tex]
Therefore, the sum of the geometric series [tex]\(-4, -16, -64, -256, \ldots\)[/tex] with [tex]\(n=6\)[/tex] is [tex]\(-5460\)[/tex].
The correct answer is:
[tex]\[
-5460
\][/tex]
