https://file.aiquickdraw.com/mathbot/file/1730486923067jnnlpvtb.jpg

To solve this problem, we need to convert the given recursive formula into an explicit formula for \( a_n \).

The recursive formula is:
\[
a_n = a_{n-1} + 15, \quad a_1 = 5
\]

This is an arithmetic sequence where each term increases by a common difference, \( d = 15 \). For an arithmetic sequence, the \( n \)-th term \( a_n \) can be expressed explicitly as:
\[
a_n = a_1 + (n - 1) \cdot d
\]

### Step-by-Step Solution
1. **Identify \( a_1 \):** From the given information, \( a_1 = 5 \).
2. **Identify the common difference \( d \):** The recursive formula shows that each term is 15 more than the previous term, so \( d = 15 \).
3. **Plug into the formula for the explicit term:**
   \[
   a_n = 5 + (n - 1) \cdot 15
   \]
4. **Simplify the expression:**
   \[
   a_n = 5 + 15n - 15
   \]
   \[
   a_n = 15n - 10
   \]

### Final Answer
The explicit formula for \( a_n \) is:
\[
a_n = 15n - 10
\]

Would you like more details on this solution or have any questions?

### Further Questions
1. What is an arithmetic sequence, and how does it differ from a geometric sequence?
2. How would the formula change if the common difference were negative?
3. Could you derive an explicit formula for a different recursive formula, say \( a_n = a_{n-1} + 20 \)?
4. How can you verify that the explicit formula matches the recursive formula by calculating a few terms?
5. What would happen to the formula if \( a_1 \) was a different number?

### Tip
When converting recursive formulas to explicit ones, check for a pattern by calculating the first few terms. This can help you understand the sequence better before applying formulas.

Write an explicit formula for the following recursive formula: a_n = a_(n-1) + 15; a_1 = 5

Learn how to convert a recursive formula to an explicit formula in an arithmetic sequence. This problem provides a step-by-step solution for the recursive formula a_n = a_(n-1) + 15 with a_1 = 5, suitable for Grade 9 algebra students.

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