Sequences and Summation Notation
A bank account that earns interest, a population counted each year, the rows in a triangle of dots: each of these is a list of numbers in a fixed order, and each number is tied to its position in line. A sequence is the tool for writing that list precisely, and summation notation is the tool for adding its terms without writing out a long string of plus signs.
By the end of this page you can:
- Read a sequence as a function whose inputs are the positive integers and find any term from its formula.
- List the first several terms of a sequence given its nth-term formula.
- Read sigma notation and evaluate a finite sum written with it.
The floor. A sequence assigns one number a_n to each positive integer n, so plugging in n gives the term in that position. Sigma notation is shorthand for a sum: the symbol below tells you where the index starts, the symbol above tells you where it stops.
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson, Precalculus 7e |
| Chapter | 12 |
| Section | 12.1 Sequences and Summation Notation |
| Open alternate | OpenStax Precalculus 2e, 11.1 Sequences and Their Notations |
| Difficulty | Core |
| Time | 30 minutes |
Before You Start
The Idea
A sequence is a function whose domain is the positive integers 1, 2, 3, and so on. Instead of writing f(n), the output is written a_n and called the nth term. To find a specific term, substitute that position number for n in the formula. The list of outputs a_1, a_2, a_3, ... is the sequence itself.
Once you have terms, you often want their sum. Writing a_1 + a_2 + a_3 + ... + a_n out in full is clumsy, so summation notation uses the Greek capital sigma. The expression to the right of sigma is the rule for a typical term, the letter below sigma is the index of summation and its starting value, and the number above sigma is the last value the index takes. You add the terms you get by stepping the index from the bottom value up to the top value.
Key Formulas
The nth term of a sequence is the value of its defining formula at position n:
\[ a_n = f(n), \quad n = 1, 2, 3, \dots \]
Sigma notation collects a finite sum of terms:
\[ \sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \cdots + a_n \]
Here k is the index of summation, 1 is the lower limit, and n is the upper limit.
Source: Stewart 7e, Section 12.1 Sequences and Summation Notation.
Worked Example
A sequence is defined by a_n = 2n + 1. Find the first four terms, then evaluate the sum of those four terms written as the sigma from k = 1 to 4 of (2k + 1).
Step 1. Find each term by substituting the position number.
- a_1 = 2(1) + 1 = 3
- a_2 = 2(2) + 1 = 5
- a_3 = 2(3) + 1 = 7
- a_4 = 2(4) + 1 = 9
Step 2. The sum from k = 1 to 4 of (2k + 1) adds exactly these four terms, since the index k runs from the lower limit 1 up to the upper limit 4.
Step 3. Add them: 3 + 5 + 7 + 9.
Show the answer
The first four terms are 3, 5, 7, and 9. Their sum is
\[ \sum_{k=1}^{4} (2k + 1) = 3 + 5 + 7 + 9 = 24. \]
A common slip is to start the index at 0 or to stop one term early. The lower limit 1 and upper limit 4 mean four terms, k = 1, 2, 3, 4, no more and no fewer.
Connections
This builds on function notation, since a sequence is just a function restricted to the positive integers, so finding a_n is the same skill as evaluating f(n). It leads next to arithmetic sequences, where each term adds a fixed amount to the one before, and to geometric sequences, where each term multiplies the one before by a fixed ratio. Both of those use the nth-term idea and sigma notation from here to write and add their terms.
Resources
- Stewart, Redlin, Watson, Precalculus 7e, Section 12.1 Sequences and Summation Notation (primary).
- OpenStax Precalculus 2e, Section 11.1 Sequences and Their Notations (alternate, free online).
Last updated: 2026-06-24