A geometric progression (GP) is a sequence of numbers where each term is obtained by multiplying the previous term by the same fixed, non-zero number called the common ratio.

Quick Scoop: Core Idea

  • In a GP, the ratio termn+1termn\frac{\text{term}_{n+1}}{\text{term}_n}termn​termn+1​​ is constant for all terms.
  • Example: 2,4,8,16,32,…2,4,8,16,32,\dots 2,4,8,16,32,… has common ratio r=2r=2r=2, since 4/2=8/4=16/8=24/2=8/4=16/8=24/2=8/4=16/8=2.
  • Another example: 10,5,2.5,1.25,…10,5,2.5,1.25,\dots 10,5,2.5,1.25,… has common ratio r=1/2r=1/2r=1/2.

General form

A geometric progression with first term aaa and common ratio rrr looks like:

a, ar, ar2, ar3, …a,;ar,;ar^2,;ar^3,;\dots a,ar,ar2,ar3,…

The nnn-th term (often written ana_nan​ or TnT_nTn​) is:

an=a rn−1a_n=a,r^{n-1}an​=arn−1

Simple example story

Imagine you start with 3 bacteria and they triple every hour.

  • Hour 0: 333
  • Hour 1: 3×3=93\times 3=93×3=9
  • Hour 2: 9×3=279\times 3=279×3=27
  • Hour 3: 27×3=8127\times 3=8127×3=81

This sequence 3,9,27,81,…3,9,27,81,\dots 3,9,27,81,… is a geometric progression with first term a=3a=3a=3 and common ratio r=3r=3r=3.

GP vs arithmetic progression (quick contrast)

  • Arithmetic progression (AP): you add the same number each time (e.g., 3,7,11,15,…3,7,11,15,\dots 3,7,11,15,…).
  • Geometric progression (GP): you multiply by the same number each time (e.g., 3,9,27,81,…3,9,27,81,\dots 3,9,27,81,…).

Both increasing and decreasing sequences can be geometric, as long as the ratio between consecutive terms stays constant.

TL;DR: A geometric progression is a sequence like a,ar,ar2,…a,ar,ar^2,\dots a,ar,ar2,… where each term is found by multiplying the previous one by the same fixed number rrr, the common ratio.

Information gathered from public forums or data available on the internet and portrayed here.