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 exampleor
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 ofThis
will mean not having to work out the first 19 terms if we want to
find the 20th 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 writeThis
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 formwith
common difference
and
Hence
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