To determine the $n$-th term of the sequence $1, 7, 13, \dots$, we need to find the pattern in the sequence.

Step 1: Check the difference between consecutive terms

• $7 - 1 = 6$
• $13 - 7 = 6$

The sequence has a common difference of $6$, indicating that it is an arithmetic sequence.

Step 2: General form of an arithmetic sequence

The $n$-th term of an arithmetic sequence is given by: $a_n = a_1 + (n-1) \cdot d$ where:

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

Step 3: Plug in the values

Here:

• $a_1 = 1$,
• $d = 6$.

The formula becomes: $a_n = 1 + (n-1) \cdot 6$

Step 4: Simplify

$a_n = 1 + 6n - 6$ $a_n = 6n - 5$

Final Answer:

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

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Would you like a step-by-step breakdown of finding the formula?
Here are 5 related questions you might find useful:

1. What is the sum of the first $n$ terms of this sequence?
2. How would the formula change if the sequence started with a different first term?
3. What is the 10th term of the sequence $1, 7, 13, \dots$?
4. How can you verify if a number belongs to this sequence?
5. Can you derive the $n$-th term of a geometric sequence?

Tip: Always check for common differences or ratios to identify the type of sequence.