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.