Summary: Arithmetic Sequences

Key Equations

recursive formula for nth term of an arithmetic sequence [latex]{a}_{n}={a}_{n - 1}+d \text{ for } n\ge 2[/latex]
explicit formula for nth term of an arithmetic sequence [latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/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