MathematicsChapter 4: Sequences and Series

Arithmetic Sequence

An arithmetic sequence (AP) has a constant difference d: a_{n+1} - a_n = d.

Learning objectives

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

Definition

An arithmetic sequence (AP) has a constant difference d: a{n+1} - an = d. First term a. General term a_n = a + (n-1)d. Meaning: you add the same jump each time (salary increase by a fixed amount). d can be negative (decreasing AP).

Mathematical form

a=3, d=5

a₇ = 3 + 6*5

a_n = d. First term a. General term a_n

d=5 a₇

Where / symbols

ARITHMETIC SEQUENCE: An arithmetic sequence (AP) has a constant difference d: a_{n+1} - a_n = d. First term a. General term a_n = a + (n-1)d. Meaning: you add the same jump each time (salary increase by a fixed amount). d can be negative (decreasing AP).
1. COMMON DIFFERENCE: d = a2-a1 = a3-a2. If the differences are not equal, it is not an AP. 3, 3, 3, ... is an AP with d=0.
2. FIND A TERM: Plug n. To find n given a_n: solve a+(n-1)d = that value. n must come out a positive integer in a 'which term' question.

Relationship

a_n = a + (n-1)d

Overview

Spot d, then the nth term:

Common Difference

d = a2-a1 = a3-a2. If the differences are not equal, it is not an AP. 3, 3, 3, ... is an AP with d=0.

Examples 5,9,13 d=4 · 10,7,4 d=-3 · Not 1,2,4,8

Find A Term

Plug n. To find n given a_n: solve a+(n-1)d = that value. n must come out a positive integer in a 'which term' question.

Examples 10th term: a+9d · Solve for n · Check by listing if small

WHAT — ARITHMETIC SEQUENCE

RULE — 1. COMMON DIFFERENCE

d = a2-a1 = a3-a2. If the differences are not equal, it is not an AP. 3, 3, 3, ... is an AP with d=0.

METHOD — 2. FIND A TERM

Plug n. To find n given a_n: solve a+(n-1)d = that value. n must come out a positive integer in a 'which term' question.

WORKED EXAMPLE

Worked: 7th term of 3, 8, 13, ... a=3, d=5 a_7 = 3 + 6*5 = 3 + 30 = 33

How to answer a arithmetic sequence 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

7th term of 3, 8, 13, ...
a=3, d=5
a₇ = 3 + 6*5
= 3 + 30 = 33

In simple words

The jump is (n-1) steps from a1, not n steps. 7th term uses 6 jumps.

Key points

common difference d
a_n = a + (n-1)d
subtract neighbours
n starts at 1

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.