Introduction to Sequences and Series
Simple Explanation
A sequence is an ordered list of numbers, each called a term, following some pattern — written a₁, a₂, a₃, ..., aₙ. A series is what you get when you add the terms of a sequence together — the sum of the first n terms is written Sₙ.
Why Do We Need It?
Distinguishing a sequence (a list) from a series (a sum) is essential — the rest of this chapter builds separate formulas for each, and mixing them up is a common early mistake.
Worked Example
Identify terms of a sequence and a partial sum
A sequence is defined by aₙ = 2n + 1. Find a₁, a₂, a₃, and the sum of these first three terms, S₃.
Why Does This Work?
A sequence rule (like aₙ=2n+1) simply defines a pattern for generating each term from its position — a series is a separate, entirely dependent operation that just adds up however many of those generated terms you choose.
Real-Life Example
Monthly savings deposits
Someone deposits a fixed pattern of amounts into a savings account each month.
The list of individual monthly deposits is a sequence; the running total in the account after several months is the corresponding series (sum).
Practice
A sequence is defined by aₙ = 3n − 1. Find a₄.
EasyCommon mistake
Using "sequence" and "series" interchangeably — a sequence is a list of terms; a series is the sum of those terms. They are related but different objects.
Quick Review
- A sequence is an ordered list of terms: a₁, a₂, a₃, ..., aₙ.
- A series is the sum of a sequence's terms, written Sₙ for the first n terms.
- A sequence rule generates terms; summing those terms produces the series.