Arithmetic Sequences and Series

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between the consecutive terms is constant.

Learning Objectives

Calculate the nth term of an arithmetic sequence and describe the properties of arithmetic sequences

Key Takeaways

Key Points

  • The behavior of the arithmetic sequence depends on the common difference [latex]d[/latex].
  • Arithmetic sequences can be finite or infinite.

Key Terms

  • arithmetic sequence: An ordered list of numbers wherein the difference between the consecutive terms is constant.
  • infinite: Boundless, endless, without end or limits; innumerable.

An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence [latex]5, 7, 9, 11, 13, \cdots[/latex] is an arithmetic sequence with common difference of [latex]2[/latex].

  • [latex]a_1[/latex]: The first term of the sequence
  • [latex]d[/latex]: The common difference of successive terms
  • [latex]a_n[/latex]: The [latex]n[/latex]th term of the sequence

The behavior of the arithmetic sequence depends on the common difference [latex]d[/latex].

If the common difference, [latex]d[/latex], is:

  • Positive, the sequence will progress towards infinity ([latex]+\infty[/latex])
  • Negative, the sequence will regress towards negative infinity ([latex]-\infty[/latex])

Note that the first term in the sequence can be thought of as [latex]a_1+0\cdot d,[/latex] the second term can be thought of as [latex]a_1+1\cdot d,[/latex] the third term can be thought of as [latex]a_1+2\cdot d, [/latex]and so the following equation gives [latex]a_n[/latex]:

[latex]a_n= ​a_1+(n−1) \cdot d[/latex]

Of course, one can always write out each term until getting the term sought—but if the 50th term is needed, doing so can be cumbersome.