Defining a Mathematical Sequence
A mathematical sequence is an ordered list of numbers, objects, or events that follows a specific pattern or rule. Each item in the list is called a term, and they are typically denoted by a subscript, like a₁, a₂, a₃, and so on. The order of the terms is crucial, distinguishing a sequence from a mere set of elements.
Key Characteristics of Sequences
Sequences can be either finite, possessing a limited number of terms, or infinite, continuing indefinitely without end. They are commonly classified as arithmetic, where each subsequent term is generated by adding a constant difference, or geometric, where each term is derived by multiplying the preceding term by a constant ratio.
Example of a Simple Sequence
Consider the sequence of positive even numbers: 2, 4, 6, 8, 10, ... In this example, the first term (a₁) is 2, the second (a₂) is 4, and so forth. The rule is to add 2 to the previous term to obtain the next. This illustrates an infinite arithmetic sequence with a common difference of 2.
Sequences vs. Series
It is essential to differentiate between a sequence and a series. A sequence is strictly an ordered list of terms, whereas a series represents the sum of those terms. For instance, if the sequence is 2, 4, 6, 8, the corresponding series would be 2 + 4 + 6 + 8 = 20, indicating the sum.