MathematicsChapter 4: Sequences and Series

Geometric Sequence and Means

A geometric sequence (GP) has a constant ratio r: a_{n+1}/a_n = r (a_n != 0).

Learning objectives

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

Definition

A geometric sequence (GP) has a constant ratio r: a{n+1}/an = r (a_n != 0). First term a. General term a_n = a r^n-1. One geometric mean between a and b (same sign, ab>0) is G = sqrt(a b) or -sqrt(ab) as the book. To insert k geometric means, r = (ba)1k+1}. Meaning: you multiply by the same factor each time (compound growth).

Mathematical form

a=3, r=2

a₅ = 3 * 2⁴

a_n = r (a_n

a_n = a r

Where / symbols

GEOMETRIC SEQUENCE AND MEANS: A geometric sequence (GP) has a constant ratio r: a_{n+1}/a_n = r (a_n != 0). First term a. General term a_n = a r^{n-1}. One geometric mean between a and b (same sign, ab>0) is G = sqrt(a b) or -sqrt(ab) as the book. To insert k geometric means, r = (b/a)^{1/(k+1)}. Meaning: you multiply by the same factor each time (compound growth).
1. GP TERMS: 2, 6, 18, 54: r=3. 8, -4, 2, -1: r=-1/2. If r=1, constant sequence. If r=-1, two-step repeat. a_n = 0 after a start of 0 is messy — school GPs usually a != 0.
2. GEOMETRIC MEANS: One GM: sqrt(ab) if a,b>0. k means: k+2 term GP from a to b. r^{k+1} = b/a. List a r, a r^2, ..., a r^k.

Relationship

a_n = a r^n-1

Overview

Spot r, then means:

Gp Terms

2, 6, 18, 54: r=3. 8, -4, 2, -1: r=-12. If r=1, constant sequence. If r=-1, two-step repeat. a_n = 0 after a start of 0 is messy — school GPs usually a != 0.

Examples r = a2/a1 · nth term a r^{n-1} · n-1 multiplies

Geometric Means

One GM: sqrt(ab) if a,b>0. k means: k+2 term GP from a to b. r^k+1 = ba. List a r, a r^2, ..., a r^k.

Examples GM of 4 and 9 is 6 · Two GMs between 3 and 24: r=2 · 3,6,12,24

WHAT — GEOMETRIC SEQUENCE AND MEANS

RULE — 1. GP TERMS

2, 6, 18, 54: r=3. 8, -4, 2, -1: r=-12. If r=1, constant sequence. If r=-1, two-step repeat. a_n = 0 after a start of 0 is messy — school GPs usually a != 0.

METHOD — 2. GEOMETRIC MEANS

One GM: sqrt(ab) if a,b>0. k means: k+2 term GP from a to b. r^k+1 = ba. List a r, a r^2, ..., a r^k.

WORKED EXAMPLE

Worked: 5th term of 3, 6, 12, ... a=3, r=2 a_5 = 3 2⁴ = 316 = 48

How to answer a geometric sequence and means 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

5th term of 3, 6, 12, ...
a=3, r=2
a₅ = 3 * 2⁴
= 3*16 = 48

In simple words

Exponent is n-1, not n. 5th term is a r^4.

Key points

common ratio r
a_n = a r^n-1
divide neighbours
root of product

Exam preparation

Exam Focus

Quick revision

Revision Zone

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