To convert the explicit rule $a_n = 6n - 20$ into a recursive rule, we need to find the relationship between the term $a_n$ and the previous term $a_{n-1}$.

Step 1: Identify the explicit rule

The given explicit formula is:

$a_n = 6n - 20$

Step 2: Find $a_1$

Substitute $n = 1$ into the explicit formula to find $a_1$:

$a_1 = 6(1) - 20 = 6 - 20 = -14$

Step 3: Write the recursive rule

Now, we need to express $a_n$ in terms of $a_{n-1}$. Notice that:

$a_n = 6n - 20 \quad \text{and} \quad a_{n-1} = 6(n-1) - 20$

Simplify $a_{n-1}$:

$a_{n-1} = 6n - 6 - 20 = 6n - 26$

Now, subtract $a_{n-1}$ from $a_n$ to find the recursive relationship:

$a_n - a_{n-1} = (6n - 20) - (6n - 26)$

Simplifying the right-hand side:

$a_n - a_{n-1} = 6n - 20 - 6n + 26 = 6$

Thus, the recursive rule is:

$a_n = a_{n-1} + 6$

Final Answer:

1. Initial condition: $a_1 = -14$
2. Recursive rule: $a_n = a_{n-1} + 6$ for $n > 1$

Would you like a more detailed explanation or have any questions on this? Here are some related questions for further exploration:

1. How can recursive sequences be useful in real-world problems?
2. How do you convert a recursive sequence back into an explicit formula?
3. What are some common types of recursive sequences?
4. How do you calculate the sum of terms in a recursive sequence?
5. How can you use mathematical induction to prove properties of recursive sequences?

Tip: Recursive rules often require identifying the base case and then establishing a relationship between consecutive terms. Be mindful of the base case when writing the recursive definition!