To find a recursive formula for the sequence $12, -1, -14, -27, \dots$, we need to identify the pattern in the sequence.

Step 1: Find the common difference

This sequence appears to be an arithmetic sequence. We can confirm this by checking the differences between consecutive terms:

• $-1 - 12 = -13$
• $-14 - (-1) = -13$
• $-27 - (-14) = -13$

So, the common difference is $-13$.

Step 2: Write the recursive formula

In an arithmetic sequence, the recursive formula is generally:

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

where $a_n$ is the $n$-th term, $a_{n-1}$ is the previous term, and $d$ is the common difference.

For our sequence:

• The first term $a_1 = 12$
• The common difference $d = -13$

Thus, the recursive formula is:

$a_n = a_{n-1} - 13 \quad \text{for} \quad n \geq 2$ with the initial condition: $a_1 = 12$

Final Answer:

$a_n = a_{n-1} - 13 \quad \text{with} \quad a_1 = 12$

Would you like further explanation on this? Or have any questions about it?

Related Questions:

1. How do you find the nth term for an arithmetic sequence using a closed formula?
2. How does the recursive formula for geometric sequences differ from arithmetic sequences?
3. Can the sequence be generalized for any other common difference besides -13?
4. What is the sum of the first 10 terms of this arithmetic sequence?
5. How can you apply this sequence to real-world problems?

Tip:

To solve recursive sequences, always start by identifying the pattern (e.g., constant difference for arithmetic, constant ratio for geometric). This will guide you in constructing the recursive formula!