Lesson 6: Arithmetic Sequences

Property

A sequence is a function whose domain is the counting numbers.
A sequence can also be seen as an ordered list of numbers and each number in the list is a term.
A sequence may have an infinite number of terms (infinite sequence) or a finite number of terms (finite sequence).
The notation $a_n$ represents the $n$th term of the sequence.

Examples

• Write the first four terms of the sequence with general term $a_n = 3n + 2$. The terms are $a_1 = 3(1)+2=5$, $a_2 = 3(2)+2=8$, $a_3 = 3(3)+2=11$, and $a_4 = 3(4)+2=14$. The sequence is $5, 8, 11, 14, \ldots$.

• Write the first four terms of the sequence with general term $a_n = (-1)^n(n+1)$. The terms are $a_1 = (-1)^1(1+1)=-2$, $a_2 = (-1)^2(2+1)=3$, $a_3 = (-1)^3(3+1)=-4$, and $a_4 = (-1)^4(4+1)=5$. The sequence is $-2, 3, -4, 5, \ldots$.

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 $d=5$.