Key Concepts & Glossary

Key Equations

recursive formula for nth term of an arithmetic sequence	[latex]{a}_{n}={a}_{n - 1}+d\phantom{\rule{1}{0ex}}n\ge 2[/latex]
explicit formula for nth term of an arithmetic sequence	[latex]\begin{array}{l}{a}_{n}={a}_{1}+d\left(n - 1\right)\end{array}[/latex]

An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
The constant between two consecutive terms is called the common difference.
The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term.
The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly.
A recursive formula for an arithmetic sequence with common difference [latex]d[/latex] is given by [latex]{a}_{n}={a}_{n - 1}+d,n\ge 2[/latex].
As with any recursive formula, the initial term of the sequence must be given.
An explicit formula for an arithmetic sequence with common difference [latex]d[/latex] is given by [latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/latex].
An explicit formula can be used to find the number of terms in a sequence.
In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}+dn[/latex].

arithmetic sequence: a sequence in which the difference between any two consecutive terms is a constant

common difference: the difference between any two consecutive terms in an arithmetic sequence