The arithmetic sequence with initial value 12 and constant difference negative 6 defines a linear pattern where each term decreases by 6. You can capture this behavior precisely by writing a recursive formula that references the previous term.
This guide explains how to express that sequence recursively, compares explicit and recursive approaches, and clarifies common implementation details for learners and practitioners.
| Term Index k | Sequence Value a_k | Recursive Dependency | Explicit Formula |
|---|---|---|---|
| 1 | 12 | Initial term | 12 |
| 2 | 6 | a_2 = a_1 - 6 | 12 - 6 |
| 3 | 0 | a_3 = a_2 - 6 | 12 - 6 × 2 |
| 4 | -6 | a_4 = a_3 - 6 | 12 - 6 × 3 |
| 5 | -12 | a_5 = a_4 - 6 | 12 - 6 × 4 |
Recursive Formula Setup for k Starting at 12
To write a recursive definition, identify the first term and the change per step. For this sequence, the first term is 12 and the common difference is -6, so each later term is the previous term minus 6.
The cleanest recursive statement uses two parts: an initial condition and a recurrence relation. The initial condition anchors the sequence at index 1, while the recurrence relation defines how to move from one term to the next.
Writing the Recursive Rule Step by Step
Translating the description "starts at 12, subtract 6 each time" into symbols helps build intuition and supports more complex sequences later. The structure is always the same: base case plus recurrence step.
Define a_k as the k-th term of the sequence, assuming k is a positive integer. Set a_1 equal to 12 as the base, then express a_k in terms of a_{k-1} for all integers k greater than 1. This mirrors how you would compute values by hand or in code.
Explicit Formula Comparison
While recursive rules define terms relative to their predecessors, an explicit formula lets you compute any term directly from its index. For arithmetic sequences, the explicit form is a linear function of k, which is easy to derive from the first term and common difference.
For this sequence, the explicit rule is a_k = 12 - 6(k - 1), which simplifies to a_k = 18 - 6k. You can verify that plugging in k = 1, 2, 3 yields the same values produced by the recursive process.
Implementation in Code and Algorithms
When implementing the sequence in software, recursive formulations must be handled carefully to avoid excessive recomputation. A simple loop or memoization can turn the recurrence into an efficient algorithm suitable for generating terms or validating properties.
Whether you use an iterative loop or a recursive function, track the current index and value explicitly. Start with index 1 and value 12, then repeatedly subtract 6 to advance, matching the arithmetic pattern precisely.
Key Takeaways for Arithmetic Sequences
- Identify the first term and the common difference to define the sequence.
- Write a base case for the smallest valid index and a recurrence relation for later terms.
- Verify consistency by checking that recursive values align with the explicit formula.
- Choose indexing that matches the context, whether starting at k = 1 or another integer.
- Use iteration or memoization in code to implement the sequence efficiently.
FAQ
Reader questions
How do I write the recursive rule for this sequence using function notation a_k?
Define a_1 = 12 and for k > 1 let a_k = a_{k-1} - 6. This captures both the starting value and the constant difference of negative 6.
Can I use k = 0 as the starting index instead of k = 1?
Yes, you can shift the indexing. If you set a_0 = 12, the recurrence becomes a_k = a_{k-1} - 6 for k >= 1, which corresponds to an explicit formula a_k = 12 - 6k.
What is the common difference in this arithmetic sequence?
The common difference is -6, because each term is exactly 6 less than the previous term, resulting in a steady decrease. Compute the first few terms using the recursion and compare them to plugging k into the explicit formula a_k = 18 - 6k; if they match for several indices, the definitions are consistent.