Lesson 12.2: Arithmetic Sequences

New Concept

An arithmetic sequence has a constant difference, $d$, between terms. We'll use this 'common difference' to find any term in the sequence using the formula $a_n = a_1 + (n-1)d$ and to calculate the sum of its first $n$ terms.

What’s next

Now, let’s apply this. You'll work through interactive examples and practice problems to master finding terms and sums of arithmetic sequences.

Property

An arithmetic sequence is a sequence where the difference between consecutive terms is always the same.

The difference between consecutive terms, $a_n - a_{n-1}$, is $d$, the common difference, for $n$ greater than or equal to two.

Examples

• The sequence $6, 11, 16, 21, 26, \ldots$ is arithmetic because the difference is always 5. The common difference is $d=5$.

Property

The general term of an arithmetic sequence with first term $a_1$ and the common difference $d$ is

Examples

• To find the 20th term of a sequence where $a_1 = 5$ and $d = 4$, use the formula: $a_{20} = 5 + (20 - 1)4 = 5 + 19 \cdot 4 = 81$.

Property

The sum, $S_n$, of the first $n$ terms of an arithmetic sequence is

where $a_1$ is the first term and $a_n$ is the $n$th term.