sequence definition math

In mathematics, a sequence is a list of numbers or objects, usually in a specific order. The individual elements in a sequence are often called terms, and the number of terms is called the length of the sequence.

Here are the steps to define a sequence:

1. Identify the first term in the sequence. This is usually denoted as a1.

2. Identify the common difference or ratio if the sequence is arithmetic or geometric, respectively. The common difference is the difference between consecutive terms in an arithmetic sequence, while the common ratio is the ratio of consecutive terms in a geometric sequence.

3. Write down the general term of the sequence. This is usually denoted as an. For an arithmetic sequence, the general term is a1 + (n-1)d, where d is the common difference. For a geometric sequence, the general term is a1 * r^(n-1), where r is the common ratio.

4. If the sequence is finite, identify the last term. If the sequence is infinite, this step is not applicable.

5. Finally, list the sequence. If the sequence is finite, list all the terms from the first to the last. If the sequence is infinite, list the first few terms followed by an ellipsis (...).

For example, the sequence {2, 4, 6, 8, 10} is an arithmetic sequence with a first term of 2 and a common difference of 2. The general term is 2 + (n-1)2 = 2n.