Geometric Sequence and Means

Geometric Sequence and Means

What is it?

A geometric sequence (GP) has a constant ratio r: a{n+1}/an = r (an != 0). First term a. General term an = 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).

In simple words

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

Important terms

GEOMETRIC SEQUENCE AND MEANS: A geometric sequence (GP) has a constant ratio r: a{n+1}/an = r (an != 0). First term a. General term an = 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).

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.

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.

What you should learn

• 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.

Examples

• r = a2/a1
• nth term a r^n-1
• n-1 multiplies

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.

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.

Full school-level explanation

A geometric sequence (GP) has a constant ratio r: a{n+1}/an = r (an != 0). First term a. General term an = 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).

Spot r, then means:

WHAT — GEOMETRIC SEQUENCE AND MEANS

A geometric sequence (GP) has a constant ratio r: a{n+1}/an = r (an != 0). First term a. General term an = 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).

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

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.

Remember

• divide neighbours

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.

Remember

• root of product

Examples

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

How to answer a geometric sequence and means question

How it works

• 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.

Remember

• common ratio r
• a_n = a r^n-1
• 48
• r to the n-1
• divide neighbours
• root of product
• common ratio r
• a_n = a r^n-1

How the ideas connect

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

Deeper understanding

Go beyond memorising Geometric Sequence and Means. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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

Common mistakes

Exam focus

Core Idea

1. GP TERMS

2. GEOMETRIC MEANS

Worked Example

Exam focus

Quick revision

• common ratio r
• a_n = a r^n-1
• 48
• r to the n-1
• GEOMETRIC SEQUENCE AND MEANS: A geometric sequence (GP) has a constant ratio r: a{n+1}/an = r (a_n != 0). First term a. General
• 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 =

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

For long answers on Geometric Sequence and Means: define clearly → explain the main parts → give one example → state one exam caution.