https://file.aiquickdraw.com/mathbot/file/17261584667915vx0zrda.jpg

The sequence provided is: \( 2, -4, 8, \dots \).

To find the equation, observe the following:

1. The first term is \( 2 \).
2. The second term is \( -4 \).
3. The third term is \( 8 \).

Notice that the signs alternate between positive and negative, and the magnitudes of the terms are powers of 2.

Looking at the absolute values:
- 1st term: \( 2 = 2^1 \)
- 2nd term: \( 4 = 2^2 \)
- 3rd term: \( 8 = 2^3 \)

Thus, the terms follow the pattern \( 2^n \), but with alternating signs. The alternating signs can be represented by \( (-1)^{n-1} \).

Therefore, the equation for the nth term of the sequence can be written as:

\[
f(n) = 2^n \cdot (-1)^{n-1}
\]

In the form provided in the image, this can be written as:

\[
f(n) = 2 \cdot (-2)^{n-1}
\]

Would you like further details or clarifications?

Here are five related questions:
1. How do you identify alternating patterns in sequences?
2. Can the alternating sign be represented in other ways?
3. How would this formula change if the first term was negative?
4. What are other common types of sequences?
5. How can you use recursion to define this sequence?

**Tip:** When dealing with alternating signs in sequences, always check for powers of \(-1\) or similar factors to simplify the pattern.

Write an equation to describe the sequence 2, -4, 8, ... using n to represent the position of a term in the sequence.

Learn how to find the equation for the sequence 2, -4, 8, ... by identifying its alternating pattern and geometric progression. This solution demonstrates the use of powers of two and alternating signs to describe the sequence.

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