Geometric Progressions

Exam code: 4037

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Geometric sequences

In a geometric progression (also called geometric sequence) there is a common ratio, r, between consecutive terms in the sequence
- For example, 2, 6, 18, 54, 162, … is a progression with the rule ‘start at two and multiply each number by three’
  - The first term, a, is 2
  - The common ratio, r, is 3
A geometric progression can be increasing (r > 1) or decreasing (0 < r < 1)
If the common ratio is a negative number the terms will alternate between positive and negative values
- For example, 1, -4, 16, -64, 256, … is a sequence with the rule ‘start at one and multiply each number by negative four’
  - The first term, a, is 1
    - The common ratio, r, is -4
Each term of a geometric progression is referred to by the letter u with a subscript determining its place in the sequence

$u_{n} = a r^{n - 1}$

Where $a$ is the first term, and $r$ is the common ratio
- This formula allows you to find any term in the geometric progression
- It is given in the list of formulas, you do not need to know how to derive it
Enter the information you have into the formula and find the value of the term
Sometimes you will be given a term and asked to find the first term or the common ratio
- Substitute the information into the formula and solve the equation
Sometimes you will be given two or more consecutive terms and asked to find both the first term and the common ratio
- Find the common ratio by dividing a term by the one before it
- Substitute this and one of the terms into the formula to find the first term
Sometimes you may be given a term and the formula for the n^th term and asked to find the value of n
- You can solve these using logarithms

A geometric series is the sum of the terms in a geometric progression
- It is often referred to as the sum of a geometric progression
- For the arithmetic sequence 1, 4, 7, 10, … the arithmetic series is 1 + 4 + 7 + 10 + …

The following formulae will let you find the sum of the first n terms of a geometric progression:

$S_{n} = \frac{a (1 - r^{n})}{1 - r}$ or $S_{n} = \frac{a (r^{n} - 1)}{r - 1}$

The first is given in the list of formulas and is more convenient if r < 1
- the second is more convenient if r > 1
A question will often give you the sum of a certain number of terms and ask you to find the value of the first term, the common ratio, or the number of terms within the sequence
- Substitute the information into the formula and solve the equation

If |r| < 1, then the sum of a geometric progression converges to a finite value given by the formula

$S_{\infty} = \frac{a}{1 - r}$

S_∞ is known as the sum to infinity
If |r| ≥ 1 the sum of a geometric progression is divergent and the sum to infinity does not exist