Final answer:
To find the recursive rule for a geometric sequence, identify the first term and the common ratio. Use the general form of the recursive rule, which relates each term to the previous one by multiplying it by the common ratio. Apply the rule starting with the first term to find subsequent terms.
Step-by-step explanation:
To find the recursive rule for a geometric sequence, you first need to identify two important elements: the first term of the sequence and the common ratio. The first term is often denoted as a1, and the common ratio is denoted as r. A recursive formula will relate each term in the sequence to the one before it.
Here is the general form of a recursive rule for a geometric sequence:
- a1 = first term,
- an = r × an-1 for n ≥ 2,
Where an is the n-th term and an-1 is the term before it.
Example:
If the first term of a geometric sequence is 5 and the common ratio is 3, the recursive rule will be:
- a1 = 5,
- an = 3 × an-1, for n ≥ 2.
To use the rule, start with the first term; then multiply by the common ratio to get the second term (a2 = 3 × 5 = 15), and continue the process to find subsequent terms. Moreover, for sequences adhering to a pattern, such as a geometric sequence, calculators like the TI-83, 83+, or 84 have functions that can generate terms following the recursive rule, providing a quick reference for students.
Relation to Other Mathematical Concepts:
Understanding recursive sequences is essential when learning about series expansions like the Binomial theorem. In a broader sense, the concept of recursion appears in various mathematical contexts and helps develop problem-solving and pattern-recognition skills.