Question

Calculate the sum of the first nine terms of a series whose first term is −4 and has a common ratio of −2 .(1 point)
Responses

2,052
2,052

−684
negative 684

−2,044
negative 2,044

684

Answer 1

To find the sum of the first nine terms of a geometric series, we can use the formula for the sum of the first \( n \) terms of a geometric series:

\[
S_n = a \frac{1 - r^n}{1 - r}
\]

where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

In this case:
- \( a = -4 \)
- \( r = -2 \)
- \( n = 9 \)

Now, we can plug in the values into the formula:

\[
S_9 = -4 \frac{1 - (-2)^9}{1 - (-2)}
\]

Calculating \( (-2)^9 \):

\[
(-2)^9 = -512
\]

Now substitute this value back into the equation:

\[
S_9 = -4 \frac{1 - (-512)}{1 + 2}
\]
\[
= -4 \frac{1 + 512}{3}
\]
\[
= -4 \frac{513}{3}
\]
\[
= -4 \cdot 171
\]
\[
= -684
\]

Thus, the sum of the first nine terms of the series is:

\[
\boxed{-684}
\]