5.1 Introduction of Sequence: A Sequence is a set of things (usually numbers) in order. A sequence is a function having domain as the positive integers. Sequences are not written as ordered pairs, or drawn as graphs, a sequence is often represented by a list of its terms starting with the first term, followed by the second, and so on.
A general sequence looks something like this:
a 1, a 2, a 3,…, a n ,
where n is an integer, and a n is the nth term of the sequence..
Example: consider the sequence:
a n = 2n = 2, 4, 6, 8, 10,…,
Where n is an integer. The formula an = 2n is called an explicit formula for the sequence. By plugging any integer n into the explicit formula for the sequence, the nth term can be found.
When a sequence is given by an explicit formula, it is easy to find the nth term of the sequence. The desired n only needs to be plugged into the equation to find the nth term. Sometimes, however, a sequence is given by listing the first four or five terms, and then an nth term must be found. In such a case the only way to find the nth term is to study the first few terms and try to find a pattern, and express it in a formula. This is a form of mathematical induction, which is essentially the process of recognizing patterns. Keep in mind that more than one sequence may share the first few terms, so there is not one right answer for a problem like this.
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