A sequence is a list of numbers with some rule to calculate each number in the sequence. The rule could be a formula for the nth term, for example
or a formula which says multiply the last term by 2 and add three to get the next term,
A sequence generated by a rule which uses each term to find the next term is called an iterative sequence. Usually we are given the first term, and some rule, so we can work out all succeeding terms. Sometimes it may be possible to find what is called a closed form for the nth term. The closed form is a formula for
in terms of n and not in terms of
This will mean not having to work out the first 19 terms if we want to find the 20 th term.
Example:
It might be helpful to write down the first few terms of the sequence, noting that the rule says multiply each term by 3 to get the next term:
1, 3, 9, 27, 81...
The sequence consists of all the powers of n, starting from the zeroth power,
Hence we can write
This is just an expression for terms in an arithmetic sequence.
Example:
It might be helpful to write down the first few terms of the sequence, noting that the rule says add 4 to each term to get the next term:
3, 7, 11, 15, 19......
This sequence is just an arithmetic sequence, of the form
with common difference
and
Hence