A sequence is an ordered list of numbers or other mathematical objects. Each element in a sequence is called a term. Sequences can be finite (ending after a certain number of terms) or infinite (continuing indefinitely). We use specific notations to describe sequences, like $a_n$ to represent the $n$-th term of the sequence. Understanding the notation helps us define and work with different types of sequences.
Sequence notation helps us express patterns and relationships within the sequence. For instance, an arithmetic sequence can be defined by a common difference between consecutive terms, while a geometric sequence can be described by a common ratio. Recognizing these patterns and representing them using proper notation is crucial for solving problems related to sequences.
Match the term with its correct definition:
| Term | Definition |
|---|---|
| 1. Sequence | A. The term in the $n$-th position |
| 2. Term | B. A function whose domain is the set of natural numbers |
| 3. Finite Sequence | C. Has a last term |
| 4. Infinite Sequence | D. A number in a sequence |
| 5. $a_n$ | E. Continues indefinitely |
Complete the following paragraph using the words provided (Term, Ordered, Sequence, Notation, Pattern).
A __________ is an __________ list of elements, often numbers. Understanding the __________ used to represent a __________ is essential for identifying the __________ in it. Each element is called a __________.
Describe a real-world situation where sequences are used. Explain how the sequence helps to solve a problem or make a prediction.
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