To convert the explicit formula $a_n = 15n - 38$ into a recursive formula, we need to express $a_n$ in terms of $a_{n-1}$.

Step 1: Write out the explicit formula for $a_n$ and $a_{n-1}$:

• $a_n = 15n - 38$
• $a_{n-1} = 15(n-1) - 38 = 15n - 15 - 38 = 15n - 53$

Step 2: Find the recursive relation.

We need to relate $a_n$ to $a_{n-1}$. Start with:

$a_n = 15n - 38$ Now express $a_n$ in terms of $a_{n-1}$:

$a_n = a_{n-1} + (15n - 38) - (15(n-1) - 38)$

Simplifying:

$a_n = a_{n-1} + (15n - 38 - 15n + 53)$ $a_n = a_{n-1} + 15$

Final Recursive Formula:

$a_n = a_{n-1} + 15$ with the initial condition $a_1 = 15(1) - 38 = -23$.

To summarize:

The recursive formula for $a_n$ is: $a_n = a_{n-1} + 15, \quad a_1 = -23$

Follow-up:

Would you like a further explanation or more details? Here are some related questions to explore:

1. How would the recursive formula change if the explicit formula was different, like $a_n = 20n - 50$?
2. How can we derive a general formula for a sequence with a linear explicit formula like $a_n = cn + d$?
3. How do we solve for the value of a specific term in a sequence using the recursive formula?
4. What is the difference between a recursive formula and an explicit formula?
5. How can we extend this recursive sequence to higher dimensions or multidimensional sequences?

Tip: When converting from explicit to recursive formulas, look at the difference between consecutive terms to establish the recursive relation.