## 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.