The sequence you provided is \(-2, -4, -8, \dots\), which appears to be a geometric sequence where each term is multiplied by a common ratio to get the next term.

Let's break this down:

1. The first term \(a_1 = -2\).
2. The common ratio \(r\) is:
   \[
   r = \frac{-4}{-2} = 2
   \]
   This means each term is multiplied by 2 to get the next term.

The formula for the \(n\)-th term of a geometric sequence is given by:
\[
a_n = a_1 \cdot r^{n-1}
\]
Substitute the values:
\[
a_{40} = -2 \cdot 2^{40-1} = -2 \cdot 2^{39}
\]

Now, calculate \(2^{39}\):
\[
2^{39} = 549,755,813,888
\]

Thus,
\[
a_{40} = -2 \cdot 549,755,813,888 = -1,099,511,627,776
\]

So, the 40th term \(a_{40}\) is \(-1,099,511,627,776\).

Do you want more details on this? Here are five related questions:

1. How do you calculate the sum of the first \(n\) terms in a geometric series?
2. What if the common ratio \(r\) was a fraction instead of 2?
3. How does changing the first term affect the sequence?
4. What happens to the terms if the common ratio is negative?
5. Can you solve for any specific term without using the geometric formula?

**Tip**: Always double-check the common ratio to ensure it's constant throughout the sequence.

Learn how to find the 40th term of the geometric sequence -2, -4, -8,... using the geometric sequence formula a_n = a_1 * r^(n-1). The solution demonstrates step-by-step calculations and highlights the importance of common ratios in sequences. Suitable for students in Grades 9-12.

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