Arithmetic Sequences
An arithmetic sequence is a sequence of numbers which increases or decreases by a constant amount each term.
We can write a formula for the nth term of an arithmetic sequence in the form
an = dn + c,
where d is the common difference.
Example 1:
{1, 5, 9, 13, 17, 21, 25, ...}
is an arithmetic sequence with common difference of 4.
A formula for the nth term of the sequence is
an = 4n – 3.
Example 2:
{12, 9, 6, 3, 0, –3, –6, ...}
is an arithmetic sequence with common difference of –3.
A formula for the nth term of this sequence is
an = –3n + 15.
Example 3:
{2, 3, 5, 8, 12, 17, 23, ...}
is not an arithmetic sequence. The difference a2 – a1 is 1, but the next difference a3 – a2 is 2.
No formula of the form
an = dn + c can be written for this sequence.
See also geometric sequences.
