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.