To determine the $n^{\text{th}}$ term rule for this arithmetic sequence:

The sequence is: 2, 9, 16, 23

Step 1: Find the common difference

The difference between consecutive terms:

• $9 - 2 = 7$
• $16 - 9 = 7$
• $23 - 16 = 7$

So, the common difference is $7$.

Step 2: Write the general formula for an arithmetic sequence

The general formula for the $n^{\text{th}}$ term is: $a_n = a + (n - 1)d$ Where:

• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Here:

• $a = 2$ (the first term),
• $d = 7$.

Substitute these values: $a_n = 2 + (n - 1)7$

Step 3: Simplify the formula

Distribute the $7$: $a_n = 2 + 7n - 7$ Combine like terms: $a_n = 7n - 5$

Final Answer:

The $n^{\text{th}}$ term rule is: $\boxed{7n - 5}$

Would you like further clarification or examples? Here are some related questions:

1. What is an arithmetic sequence, and how is it different from a geometric sequence?
2. How do you find the common difference in an arithmetic sequence?
3. What is the formula for the sum of the first $n$-terms of an arithmetic sequence?
4. How can you determine if a sequence is arithmetic or not?
5. Can you use this $n^{\text{th}}$-term rule to find the $10^{\text{th}}$ term?

Tip: Always check your sequence by substituting values of $n$ back into the formula to verify its correctness.