my teacher asked me to make my own rule in arithmetic sequence

Creating your own rule for an arithmetic sequence is a great way to understand the concept better. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference.

### Steps to Create Your Own Rule:

1. **Choose the First Term (\(a_1\)):** Decide on the first term of your sequence. This can be any number.
2. **Decide on the Common Difference (\(d\)):** Choose the common difference, which is the amount you will add (or subtract) to get from one term to the next.
3. **Write the General Formula:** The \(n\)th term (\(a_n\)) of an arithmetic sequence can be expressed using the formula:
   \[
   a_n = a_1 + (n - 1) \cdot d
   \]
   Here, \(n\) is the position of the term in the sequence.

### Example:

1. **First Term (\(a_1\))**: Let's say \(a_1 = 3\).
2. **Common Difference (\(d\))**: Let's choose \(d = 5\).

Using the formula:
\[
a_n = 3 + (n - 1) \cdot 5
\]

Let's calculate the first few terms using this rule:
- \(a_1 = 3 + (1 - 1) \cdot 5 = 3\)
- \(a_2 = 3 + (2 - 1) \cdot 5 = 3 + 5 = 8\)
- \(a_3 = 3 + (3 - 1) \cdot 5 = 3 + 10 = 13\)
- \(a_4 = 3 + (4 - 1) \cdot 5 = 3 + 15 = 18\)

So, the sequence is: 3, 8, 13, 18, ...

### Your Turn:

Now, try creating your own rule by picking your first term and common difference. Use the general formula to find the first few terms of your sequence.

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Do you want details or have any questions?

### Relative Questions:

1. What is the definition of an arithmetic sequence?
2. How do you determine the common difference in an arithmetic sequence?
3. Can the common difference in an arithmetic sequence be negative?
4. How would you write the general formula for an arithmetic sequence starting with a different first term?
5. How do you find the 10th term of an arithmetic sequence if you know the first term and the common difference?
6. What happens to the terms of an arithmetic sequence if the common difference is zero?
7. How do you derive the formula for the sum of the first \(n\) terms of an arithmetic sequence?
8. Can an arithmetic sequence have non-integer terms? Give an example.

### Tip:
Always double-check your common difference when calculating terms to ensure accuracy in your arithmetic sequence.

Learn how to create your own rule in arithmetic sequences with a step-by-step guide. Understand the concept of arithmetic sequences, choose the first term and common difference, and use the nth term formula to generate terms. Suitable for Grades 7-9 students learning arithmetic sequences.

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