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.