Recursive formulas are a way to define a sequence by relating each term to the previous term(s). Instead of having a direct formula to calculate the $n$th term, you need to know the term(s) before it. Typically, a recursive formula will have two parts: a starting value (or values) and a recurrence relation that shows how to get the next term from the previous one(s). It's like a set of instructions on how to build the sequence step by step!
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Sequence | A. The term that comes immediately before a given term in a sequence. |
| 2. Term | B. A function whose domain is the set of positive integers. |
| 3. Initial Value | C. The value(s) of the first term(s) in a sequence, which serve as the starting point for a recursive formula. |
| 4. Previous Term | D. A formula that defines each term of a sequence based on the preceding term(s). |
| 5. Recurrence Relation | E. Each individual number or element in a sequence. |
A __________ formula defines a sequence by relating each term to the term(s) that come before it. You need to know the __________ value(s) to start. The __________ relation shows how to calculate the next term using the previous one(s).
Consider the Fibonacci sequence, which starts with 1, 1, 2, 3, 5, ... How would you describe the Fibonacci sequence using a recursive formula? Explain why it's necessary to have two initial values in this case.
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