Definition: An arithmetic sequence is an ordered list of numbers in which each term is found by adding the same fixed number, the common difference, to the term before it.
How does an arithmetic sequence behave?
The gap between neighbors never changes. The list 7, 12, 17, 22 grows by 5 each time. If the difference is negative, the list falls, as in 40, 33, 26.
What formula gives any term?
With first term a1 and common difference d, the nth term is an = a1 + (n – 1)d. The (n – 1) appears because the first term needs no steps.
Example. Find the 30th term of 11, 15, 19, … Here a1 = 11 and d = 4. So a30 = 11 + 29(4) = 11 + 116 = 127.
Where do students slip?
They multiply by n instead of n – 1, which gives 131 above. They also mix up arithmetic and geometric sequences. To tell them apart, subtract consecutive terms: equal differences mean arithmetic. Graphing the terms gives points that fall on a straight line, so an arithmetic sequence is a linear function whose domain is the counting numbers.