To convert an explicit formula to a recursive formula, we need to identify how each term in the sequence relates to the previous terms. Since you haven’t provided the specific explicit formula, I’ll guide you through the general process:

Steps to Identify a Recursive Formula from an Explicit Formula:

1. Start with the explicit formula: The explicit formula gives the $n$-th term directly. For example: $a_n = f(n)$ where $f(n)$ is some function of $n$.

2. Express $a_{n+1}$: The recursive formula gives the next term in the sequence in terms of the previous term or terms. So, to find this, express $a_{n+1}$ by plugging $n+1$ into the explicit formula. This gives: $a_{n+1} = f(n+1)$

3. Find the relationship: Look at the relationship between $a_{n+1}$ and $a_n$ by subtracting, dividing, or otherwise manipulating the two expressions to isolate $a_{n+1}$ in terms of $a_n$.

Example

Suppose the explicit formula is: $a_n = 3n + 2$

• Step 1: Start with $a_n = 3n + 2$.
• Step 2: Find $a_{n+1}$: $a_{n+1} = 3(n+1) + 2 = 3n + 3 + 2 = 3n + 5$
• Step 3: Compare $a_{n+1}$ and $a_n$: $a_{n+1} = a_n + 3$ So the recursive formula is: $a_{n+1} = a_n + 3$

Now, if you can provide the specific explicit formula you're working with, I can find the exact recursive formula for you.

Would you like more details or have any questions?

Related Questions:

1. What is the difference between an explicit and recursive formula?
2. How do you convert a recursive formula back into an explicit formula?
3. How would the recursive formula change if the explicit formula were quadratic?
4. Can a sequence have both a recursive and an explicit formula?
5. How can you determine the initial condition for a recursive formula?

Tip: Recursive formulas often require an initial value, like $a_0$, to fully define the sequence.