Explain what an explicit formula is according to arithmetic sequence

An explicit formula in the context of an arithmetic sequence is a mathematical expression that allows you to calculate the \(n\)th term of the sequence directly, without needing to know the preceding terms. This formula defines the \(n\)th term, \(a_n\), as a function of \(n\), the position of the term in the sequence.

For an arithmetic sequence, where the difference between consecutive terms is constant, the explicit formula is given by:

\[
a_n = a_1 + (n - 1) \cdot d
\]

- \(a_n\) is the \(n\)th term of the sequence.
- \(a_1\) is the first term of the sequence.
- \(d\) is the common difference between consecutive terms.
- \(n\) is the position of the term in the sequence.

### Example
Consider an arithmetic sequence where the first term \(a_1\) is 3 and the common difference \(d\) is 5. The explicit formula for this sequence would be:

\[
a_n = 3 + (n - 1) \cdot 5 = 5n - 2
\]

Using this formula, you can find any term in the sequence. For example, the 4th term (\(n = 4\)) would be:

\[
a_4 = 5 \times 4 - 2 = 20 - 2 = 18
\]

**Would you like more details or have any questions?**

### Related Questions:
1. How does an explicit formula differ from a recursive formula in arithmetic sequences?
2. Can you provide an example of an arithmetic sequence and derive its explicit formula?
3. How do you derive the explicit formula for an arithmetic sequence from a given set of terms?
4. What are the practical applications of using explicit formulas in arithmetic sequences?
5. How does changing the common difference \(d\) affect the explicit formula and the sequence?

**Tip:** Always identify the first term and common difference before attempting to write the explicit formula for an arithmetic sequence.

Explore the explicit formula in arithmetic sequences, defining the nth term directly using the sequence's first term and common difference. Includes examples and applications. Suitable for Grades 10-12.

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