## Key Equations

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

## Key Concepts

• 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 $d$ is given by ${a}_{n}={a}_{n - 1}+d,n\ge 2$.
• As with any recursive formula, the initial term of the sequence must be given.
• An explicit formula for an arithmetic sequence with common difference $d$ is given by ${a}_{n}={a}_{1}+d\left(n - 1\right)$.
• 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 ${a}_{n}={a}_{0}+dn$.

## Glossary

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