MathematicsChapter 4: Sequences and Series

Sequences and the General Term

A sequence is a list a1, a2, a3, ...

Learning objectives

State the main ideas and notation used in sequences and the general term.
Apply the rules of sequences and the general term in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

A sequence is a list a1, a2, a3, ... with a rule for the nth term a_n. Finite: stops. Infinite: goes on. A series is the sum a1+a2+... (adding the terms). Meaning: sequence is the list; series is the running total. Always write the first few terms to check a formula.

Mathematical form

n=1: 0

n=2: 3

n=3: 8

n=4: 15

Where / symbols

SEQUENCES AND THE GENERAL TERM: A sequence is a list a1, a2, a3, ... with a rule for the nth term a_n. Finite: stops. Infinite: goes on. A series is the sum a1+a2+... (adding the terms). Meaning: sequence is the list; series is the running total. Always write the first few terms to check a formula.
1. SEQUENCE: Example: 2, 5, 8, 11, ... might be a_n = 3n-1. Index n usually starts at 1 in school. Recurrence: a_{n+1} from a_n (later AP/GP).
2. SERIES: S_n = a1+...+an. Sigma notation named: sum_{k=1}^{n} a_k. Do not mix a_n (one term) with S_n (a total).

Relationship

sequence = list
series = sum of the list

Overview

List vs sum, then a_n:

Sequence

Example: 2, 5, 8, 11, ... might be an = 3n-1. Index n usually starts at 1 in school. Recurrence: a{n+1} from a_n (later AP/GP).

Examples a1 first term · Write 4 terms · n is a positive integer

Series

Sn = a1+...+an. Sigma notation named: sumk=1ⁿ ak. Do not mix an (one term) with S_n (a total).

Examples S_1 = a1 · S_n - S_{n-1} = a_n · Infinite sum later GP

WHAT — SEQUENCES AND THE GENERAL TERM

RULE — 1. SEQUENCE

Example: 2, 5, 8, 11, ... might be an = 3n-1. Index n usually starts at 1 in school. Recurrence: a{n+1} from a_n (later AP/GP).

METHOD — 2. SERIES

Sn = a1+...+an. Sigma notation named: sumk=1ⁿ ak. Do not mix an (one term) with S_n (a total).

WORKED EXAMPLE

Worked: first four terms of a_n = n^2 - 1. n=1: 0 n=2: 3 n=3: 8 n=4: 15

How to answer a sequences and the general term question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

first four terms of a_n = n^2 - 1.
n=1: 0
n=2: 3
n=3: 8
n=4: 15

In simple words

an is one term. Sn is a sum. Do not call S_n a sequence term.

Key points

sequence = list
series = sum of the list
nth term
S_n is a sum

Exam preparation

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