A sequence is an ordered list of numbers or mathematical objects. The key characteristic is that the order of elements matters. For example, the sequence two, four, six, eight is different from four, two, eight, six. Sequences can be finite, with a specific number of terms, or infinite, continuing without end.
There are two main types of sequences. A finite sequence has a specific number of terms and ends. For example, the sequence one, three, five, seven, nine has exactly five terms. An infinite sequence continues without end, like the powers of two: two, four, eight, sixteen, and so on forever.
Sequences use subscript notation to identify terms. We write a sub n to represent the nth term of a sequence, where n is the index or position. For example, in the sequence two, four, six, eight, a sub one equals two, a sub two equals four, and so on. We can often express sequences with a general formula, like a sub n equals two n for this example.
Two common types of sequences are arithmetic and geometric sequences. In an arithmetic sequence, each term increases by a constant difference. For example, three, seven, eleven, fifteen, where we add four each time. In a geometric sequence, each term is multiplied by a constant ratio. For example, two, six, eighteen, fifty-four, where we multiply by three each time.
To summarize what we have learned about sequences: A sequence is an ordered list of numbers where order matters. Sequences can be either finite with a specific number of terms, or infinite continuing forever. Common types include arithmetic sequences with constant differences and geometric sequences with constant ratios. Understanding sequences is fundamental to many areas of mathematics.