Which explicit formula defines the given arithmetic sequence? (General form: an = a1 + d(n − 1)
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To determine the explicit formula for an arithmetic sequence, you need two key pieces of information:

1. **The first term** of the sequence, \( a_1 \).
2. **The common difference** between consecutive terms, \( d \).

Given the general form \( a_n = a_1 + d(n − 1) \), let's define how to find each element:

- **\( a_1 \)** is simply the first term of the sequence.
- **\( d \)** is the common difference, which you find by subtracting any term from the previous term in the sequence.

### Example:
If you are given a sequence like 3, 7, 11, 15,...

1. **Identify \( a_1 \):** The first term is 3.
2. **Determine \( d \):** The difference between each term is \( 7 - 3 = 4 \).

So, the explicit formula for this sequence would be:
\[ a_n = 3 + 4(n − 1) \]

Which can be simplified to:
\[ a_n = 4n - 1 \]

Would you like to provide a specific sequence for a more detailed explanation?

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**Relative Questions:**
1. How do you calculate the common difference in an arithmetic sequence?
2. What happens to the explicit formula if the common difference is negative?
3. Can the first term \( a_1 \) be zero in an arithmetic sequence?
4. How do you find the \( n \)-th term if only the first few terms are given?
5. What is the impact on the sequence if the common difference \( d \) changes after a certain term?

**Tip:** Always check the consistency of the common difference across the sequence to confirm it's arithmetic.

Discover how to determine the explicit formula of an arithmetic sequence using the formula an = a1 + d(n - 1). Explore examples and detailed explanations on calculating the first term, common difference, and deriving the nth term. Suitable for students in Grades 9-12.

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