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:
\[ \boxed{-684} \]
1 answer