To find the expression for the given sequence, we need to identify its pattern.

The sequence is:

$-64, -128, -192, -256, \dots$

Step 1: Identify the common difference

To recognize the pattern, we look at the difference between consecutive terms:

$-128 - (-64) = -64$ $-192 - (-128) = -64$ $-256 - (-192) = -64$

Since the common difference between consecutive terms is constant ($-64$), this is an arithmetic sequence.

Step 2: General formula for an arithmetic sequence

The general formula for the $n$-th term of an arithmetic sequence is:

$a_n = a_1 + (n - 1) \cdot d$

Where:

• $a_1$ is the first term (which is $-64$),
• $d$ is the common difference (which is $-64$).

Step 3: Substitute the values into the formula

Substitute $a_1 = -64$ and $d = -64$ into the formula:

$a_n = -64 + (n - 1) \cdot (-64)$

Simplify the expression:

$a_n = -64 + (-64n + 64)$

$a_n = -64n$

Thus, the expression for the $n$-th term of the sequence is:

$a_n = -64n$

Would you like a more detailed explanation or have any questions about this? Here's some further exploration:

1. How to identify the first term and common difference in arithmetic sequences.
2. Can this formula apply to geometric sequences as well?
3. How can we find the sum of the first $n$ terms of this sequence?
4. What happens if the common difference is positive instead of negative?
5. What is the 10th term in this sequence?

Tip: Always check the common difference or ratio in a sequence first, as it helps quickly determine if the sequence is arithmetic or geometric.