The given sequence is:

$-4, 16, -64, 256, \dots$

This is a geometric sequence, where each term is obtained by multiplying the previous term by a constant ratio. To determine the common ratio ($r$):

$r = \frac{16}{-4} = -4$

Thus, the common ratio is $r = -4$.

The general formula for the sum of the first $n$ terms of a geometric sequence is:

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

where:

• $S_n$ is the sum of the first $n$ terms,
• $a$ is the first term of the sequence,
• $r$ is the common ratio, and
• $n$ is the number of terms.

Here, $a = -4$, $r = -4$, and $n = 12$.

Now, we can substitute these values into the formula:

$S_{12} = -4 \cdot \frac{1 - (-4)^{12}}{1 - (-4)}$

First, let's calculate $(-4)^{12}$:

$(-4)^{12} = 16777216$

Now, substitute this back into the formula:

$S_{12} = -4 \cdot \frac{1 - 16777216}{1 + 4}$ $S_{12} = -4 \cdot \frac{-16777215}{5}$ $S_{12} = -4 \cdot (-3355443)$ $S_{12} = 13421772$

Thus, the sum of the first 12 terms of the sequence is:

$S_{12} = 13,421,772$

Would you like further details or have any questions?

Here are 5 related questions:

1. What happens if the common ratio was positive instead of negative?
2. How would you find the sum of an infinite geometric series?
3. Can you derive the general term for this geometric sequence?
4. What is the effect of the common ratio on the convergence of the series?
5. How do geometric progressions differ from arithmetic progressions?

Tip: In a geometric progression, if the common ratio's absolute value is less than 1, the series converges; otherwise, it diverges.