# Geometric Progressions — Formula, Sum, Examples

**TL;DR**

A geometric progression is a sequence in which every term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio. This article covers the nnnth-term and sum formulas, three worked examples at Quick/Standard/Stretch tiers, the infinite-sum case, and the 1859 Australian rabbit explosion that turned a geometric progression into a national crisis.

**Last updated on June 1, 2026** | 9 min read

## A Pattern That Doubles, Triples, or Halves Forever

Some sequences add a fixed amount each step. Others _multiply_ by a fixed amount. The second kind grows — or shrinks — much faster, and that single difference reshapes how the sequence behaves.

The geometric progression 2, 4, 8, 16, … reaches 256 by term 8. The arithmetic progression 2, 4, 6, 8, … reaches 16.

## What a Geometric Progression Is

A **geometric progression** (GP), also called a geometric sequence, is a list of numbers \(a_1, a_2, a_3, …\) such that the ratio between any two consecutive terms is the same constant \(r\):  
\(\frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \cdots = r.\)  
The first term is \(a\) (sometimes \(a_1\)). The general term is  
\(a_n = a \cdot r^{n-1}.\)  
**Examples:** 
- \(2, 6, 18, 54, …\) (\(a = 2, r = 3\))  
- \(10, 5, 2.5, 1.25, …\) (\(a = 10, r = \frac{1}{2}\))  
- \(1, -3, 9, -27, …\) (\(a = 1, r = -3\))

> **Quick facts.**
> - **General term:** \(a_n = a \cdot r^{n-1}\)
> - **Common ratio:** \(r = \frac{a_{n+1}}{a_n}\)
> - **Finite sum:** \(S_n = \frac{a(1 - r^n)}{1 - r}, \quad r \neq 1.\)
> - **Infinite sum:** \(S_\infty = \frac{a}{1 - r}, \quad |r| < 1.\)
> - **Growth behaviour:** exponential if \(|r| > 1\), decay if \(0 < r < 1\), oscillation if \(r < 0\).

## The nth Term Formula

To find any specific term of a GP, multiply the first term by \(r\) as many times as needed.  
\(a_n = a \cdot r^{n-1}.\)

**Example:** Find the 7th term of \(3, 6, 12, 24, …\):  
a = 3, r = 2, n = 7.
\(a_7 = 3 \cdot 2^{6} = 192.\)

## The Sum of a Finite GP

The sum of the first n terms is  
\(S_n = \frac{a(1 - r^n)}{1 - r}, r \neq 1.\)

**Derivation:** Let \(S_n = a + ar + ar^2 + \cdots + ar^{n-1}\).

Multiply by \(r\):  
\(rS_n = ar + ar^2 + \cdots + ar^n.\)

Subtract:  
\(S_n - rS_n = a - ar^n\)  
\(S_n(1 - r) = a(1 - r^n).\)

## The Sum of an Infinite GP

When \(|r| < 1\), the finite sum formula simplifies to  
\(S_\infty = \frac{a}{1 - r}.\)

**Example:**  
\(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = \frac{1}{1 - \frac{1}{2}} = 2.\)

## Three Worked Examples of Geometric Progression

### **Quick:** Find the 5th term of the GP \(2, 6, 18, 54,…\):  
\(a = 2, r = 3, n = 5.\)
\(a_5 = 2 \cdot 3^{4} = 162.\)

### **Standard:** Find the sum of the first 8 terms of \(5, 10, 20, 40,…\).

The correct way:  
\(S_8 = \frac{5(2^8 - 1)}{2 - 1} = 1275.\)

### **Stretch:** A ball is dropped from 32 metres. Each bounce reaches \(\frac{3}{4}\) of the previous height:

Sum of downward distances:  
\(S_{\downarrow} = \frac{32}{1 - \frac{3}{4}} = 128 m.\)  
Sum of upward distances:  
\(S_{\uparrow} = \frac{24}{1 - \frac{3}{4}} = 96 m.\)  
Total:  
\(128 + 96 = 224 m.\)

## Where Geometric Progressions Show Up in the Real World

- **Compound interest:** A principal \(P\) at rate \(r\) compounded annually for \(n\) years grows as \(P(1+r)^n.\)
- **Population growth:** Before limits, populations multiply each generation.
- **Radioactive decay:** Half-life shrinks by half over iterations.
- **Music intervals:** Frequencies form a GP with common ratio \(2^{1/12}.\)

## Geometric Progression: Mistakes Students Make Most Often

1. **Confusing the common ratio with the common difference.**  
   Correct way: use multiplication, not addition.
2. **Using the wrong sum formula.**  
   Apply the correct finite or infinite sum formulas based on \(r\).
3. **Forgetting the \(r=1\) exception.**  
   Handle separately without dividing by zero.
4. **Computing infinite sums when \(|r| \geq 1\).**  
   Infinite sums diverge in these cases.

## Conclusion

- A **geometric progression** has terms that multiply by a common ratio \(r\).
- Common mistakes often involve using incorrect formulas.
- GPs are prevalent in finance, biology, and more.