Skip to lesson content

Lesson 6 of 8

Predicting What Comes Next: Exploring Sequences and Progressions · Lesson 6 of 8

Geometric Progressions

“Geometric progressions multiply the drama one term at a time.”

Learning Objectives

• Identify geometric progressions. • Find the common ratio. • Derive and use the nth-term formula. • Write recursive rules for GPs. • Compare geometric growth with arithmetic growth.

Introduction

Arithmetic progressions and geometric progressions both describe patterns in which numbers change in a regular way, but the rule behind the change is different. In an arithmetic progression, we move from one term to the next by adding or subtracting the same number each time. This fixed amount is called the common difference.

In a geometric progression, we move from one term to the next by multiplying or dividing by the same number each time. This fixed multiplier is called the common ratio. For example, in the sequence 3, 6, 12, 24, 48, … each term is obtained by multiplying the previous term by 2. So, unlike an arithmetic progression where the same amount is added repeatedly, a geometric progression changes through repeated multiplication.

Definition
Geometric Progression

A sequence in which each term after the first is obtained by multiplying the previous term by the same fixed number.

Definition
Common Ratio

The fixed multiplier between consecutive terms of a geometric progression.

Common ratioLaTeX

Deriving the nth Term

If the first term is a, then the terms are a, ar, ar², ar³,… . By the nth position, r has been multiplied n−1 times.

nth term of a GPLaTeX
Recursive GP ruleLaTeX
Growing GP

Problem
Find the nth term of 2,10,50,250,… .

  1. 1.Each term is multiplied by 5, so r=5.
  2. 2.a=2.
  3. 3.Therefore tₙ=2×5ⁿ⁻¹.
Alternating GP

Problem
Is 1,−1,1,−1,… a GP?

  1. 1.Each term is obtained by multiplying the previous term by −1.
  2. 2.So r=−1.
  3. 3.Yes, it is a GP.
Shrinking GP

Problem
For 5,15/4,45/16,135/64,… find r.

  1. 1.(15/4)÷5=3/4.
  2. 2.(45/16)÷(15/4)=3/4.
  3. 3.The ratio stays constant.
  4. 4.So r=3/4 and the sequence is a GP.

AP vs GP

Arithmetic progressionGeometric progression
Same amount addedSame factor multiplied
Constant difference dConstant ratio r
tₙ=a+(n−1)dtₙ=arⁿ⁻¹
Plots form a straight-line patternPlots generally do not lie on a straight line
AP versus GP growth The arithmetic progression 3, 6, 9, 12 forms points on a straight line, while the geometric progression 3, 6, 12, 24 rises increasingly fast. AP Versus GP Growth Equal addition produces linear growth; equal multiplication produces accelerating growth Arithmetic Progression 3, 6, 9, 12 Add 3 each time: +3, +3, +3 1 2 3 4 3 6 9 12 Term number, n Term value 3 6 9 12 Geometric Progression 3, 6, 12, 24 Multiply by 2: ×2, ×2, ×2 1 2 3 4 3 6 12 18 24 Term number, n Term value 3 6 12 24 AP Constant addition produces steady, straight-line growth. GP Constant multiplication produces increasingly faster growth.
AP versus GP growth

Practice Problems

Practice Problems
  1. Check whether 3,12,48,192,… is a GP and find r.
  2. Find the nth term of 5,25,125,… .
  3. Find the 8th term of a GP with a=3 and r=2.
  4. Check whether 16,8,4,2,… is a GP and find its nth term.
  5. Write an explicit and recursive rule for 4,12,36,108,… .
  6. Explain the difference between 'add 4 each time' and 'multiply by 4 each time'.

Key Takeaways

Key Takeaways

• GPs have a constant ratio. • The common ratio can be greater than 1, between 0 and 1, or negative. • tₙ=arⁿ⁻¹. • GP growth comes from repeated multiplication. • APs and GPs behave very differently even when their first terms look similar.

Coming Next

Next, we use geometric progressions to understand fractals and bouncing-ball patterns.

Previous · Lesson 5

Sum of the First n Natural Numbers

Next · Lesson 7

Geometric Progressions in Fractals and Bounces