The arithmetic sequence shown is:

$1, 7, 13, 19, 25, \dots$

To find the $n$-th term rule, we proceed as follows:

Step 1: Find the common difference ($d$)

Subtract any two consecutive terms: $d = 7 - 1 = 6, \quad d = 13 - 7 = 6$ Thus, the common difference is $d = 6$.

Step 2: General form of an arithmetic sequence

The formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$ where:

• $a_1$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Step 3: Substitute the known values

Here:

• $a_1 = 1$,
• $d = 6$.

Substitute into the formula: $a_n = 1 + (n-1)(6)$

Step 4: Simplify

Distribute the $6$: $a_n = 1 + 6n - 6$ $a_n = 6n - 5$

Final Answer:

The $n$-th term rule for the sequence is: $a_n = 6n - 5$

Would you like me to explain any part of this further?

Related Questions:

1. How do you find the common difference in an arithmetic sequence?
2. How can you verify that $a_n = 6n - 5$ is correct for the given sequence?
3. What are the first five terms of the sequence defined by $a_n = 6n - 5$?
4. How would the $n$-th term formula change if the first term were different?
5. Can you derive the sum of the first $n$ terms for this sequence?

Tip:

Always verify your formula by plugging in small values of $n$ (e.g., $n = 1, 2, 3$) to see if it matches the terms of the sequence.