Lesson 1: Defining and Using Sequences and Series

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

The general term of the sequence is found from the formula for writing the $n$th term of the sequence.
The $n$th term of the sequence, $a_n$, is the term in the $n$th position where $n$ is a value in the domain.

Examples

• Find a general term for the sequence $5, 10, 15, 20, 25, \ldots$. Each term is 5 times its position number, $n$. So, the general term is $a_n = 5n$.

• Find a general term for the sequence $1, -2, 3, -4, 5, \ldots$. The numbers are the position, $n$, but the signs alternate, starting with positive. So, the general term is $a_n = (-1)^{n+1}n$.