sequence

Set of elements, arranged in order according to some rule. See arithmetic progression, geometric progression, and series.

sequence

An example of a sequence from Handel's Acis and Galatea.

In music, a device allowing key modulation favoured by early keyboard composers in which a phrase is repeated sequentially, each time transposing to a different key.

sequence

To find the rule for extending the pattern in a number sequence, state the action to reach each previous term for the next term.

List of numbers that are generated by a rule. Each number in the sequence is a term. For example, 3, 7, 11, 15, 19 is a sequence of five terms generated by adding 4 to the previous term. To generate a sequence, it is necessary to know the first term, the number of terms in the sequence, and the rule.

Examples of special sequences include:

odd numbers 1, 3, 5, 7, 9... the rule is add 2

even numbers 2, 4, 6, 8, 10... the rule is add 2

square numbers 1, 4, 9, 16, 25... the rule is 12, 22, 32, and so on

triangle numbers 1, 3, 6, 10, 15... the rule is 1, 1 + 2, 1 + 2 + 3, and so on

Finding the formula The rule for a sequence only relates one term to the next term in the sequence. In order to find any term (the nth term, where n ≥ 1) in the sequence, it is necessary to find a formula that relates the term to its position in the sequence. This can be done by finding the common difference between the terms.

For example, to find the formula for the 20th term in the sequence 3, 7, 11, 15, 19..., a table can be drawn up:

The formula in this case for the nth term is 4 × the position number − 1, which can be written as 4n − 1, where n stands for the position number. Using the formula, the 20th term will be 4 × 20 − 1 = 79.

See also arithmetic progression, or arithmetic sequence, and geometric progression.