# Summary: Arithmetic Sequences

## 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] |

## 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 [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].

## 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

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