22nd term: –2, –5, –8, –11, –14, ...

To find the 22nd term in the sequence \(-2, -5, -8, -11, -14, \dots\), we first need to determine if it is an arithmetic sequence.

### Step 1: Verify if it's an Arithmetic Sequence
An arithmetic sequence has a common difference (\(d\)) between consecutive terms.

Let's calculate the common difference:
\[
d = -5 - (-2) = -5 + 2 = -3
\]
\[
d = -8 - (-5) = -8 + 5 = -3
\]
\[
d = -11 - (-8) = -11 + 8 = -3
\]
So, the common difference \(d\) is \(-3\).

### Step 2: Use the Formula for the \(n\)th Term
The formula for the \(n\)th term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n - 1) \times d
\]
Where:
- \(a_n\) is the \(n\)th term,
- \(a_1 = -2\) (the first term),
- \(d = -3\) (the common difference),
- \(n = 22\).

### Step 3: Calculate the 22nd Term
Substitute the values into the formula:
\[
a_{22} = -2 + (22 - 1) \times (-3)
\]
\[
a_{22} = -2 + 21 \times (-3)
\]
\[
a_{22} = -2 - 63
\]
\[
a_{22} = -65
\]

### Final Answer
The 22nd term of the sequence is \(-65\).

Would you like more details on this, or have any questions?

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Here are five related questions:

1. What is the sum of the first 22 terms of this sequence?
2. How would the sequence change if the common difference was \(-4\) instead of \(-3\)?
3. Can you derive the formula for the \(n\)th term of this arithmetic sequence?
4. What is the 50th term of the sequence?
5. How many terms are needed for the sequence to reach \(-100\)?

**Tip:** Always double-check your common difference when working with arithmetic sequences.

Learn how to find the 22nd term in the arithmetic sequence -2, -5, -8, -11, -14, ... using the formula for the nth term of an arithmetic sequence. Detailed step-by-step solution included. Suitable for students in Grades 7-9.

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