https://file.aiquickdraw.com/mathbot/file/1725893183985vmtk95w7.jpg

The sequence shown in the image is: 
\[
-16, -12, -8, -4, \dots
\]
This appears to be an arithmetic sequence where each term increases by a common difference. Let's analyze the problem.

### Step 1: Identify the common difference
To find the common difference \(d\), subtract the first term from the second:
\[
d = -12 - (-16) = 4
\]

### Step 2: General formula for the \(n\)-th term of an arithmetic sequence
The formula for the \(n\)-th term \(a_n\) of an arithmetic sequence is:
\[
a_n = a_1 + (n-1) \cdot d
\]
where:
- \(a_1\) is the first term,
- \(d\) is the common difference, and
- \(n\) is the term number.

### Step 3: Apply the formula to find \(a_{31}\)
Here:
- \(a_1 = -16\),
- \(d = 4\),
- \(n = 31\).

Substituting into the formula:
\[
a_{31} = -16 + (31 - 1) \cdot 4
\]
\[
a_{31} = -16 + 30 \cdot 4
\]
\[
a_{31} = -16 + 120 = 104
\]

Thus, the 31st term \(a_{31}\) is \(104\).

Would you like more details on the steps? Here are some related questions:

1. How do you derive the formula for the \(n\)-th term in an arithmetic sequence?
2. What would the sum of the first 31 terms be in this sequence?
3. How can you check whether a given sequence is arithmetic or not?
4. What is the difference between arithmetic and geometric sequences?
5. How would you solve for the common difference if it weren't immediately obvious?

**Tip:** In arithmetic sequences, the common difference remains constant between consecutive terms, making it easy to spot patterns and predict future terms.

Learn how to find the 31st term of an arithmetic sequence that begins at -16 and progresses by adding 4. This guide covers the concept of arithmetic sequences, uses the formula a_n = a_1 + (n-1) · d, and provides a step-by-step solution. Suitable for middle school students studying sequences.

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