IB Maths AA 1.3 Notes

Geometric sequences & the common ratio

A geometric sequence is a sequence in which each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio, written as $r$.

For example, in the sequence $2,6,18,54,162,\dots$, each term is found by multiplying the previous term by $3$, so the common ratio is $r=3$. So a geometric sequence has the form:

$u_1,u_1r,u_1r^2,u_1r^3,\dots$

where $u_1$ is the first term. A geometric sequence is sometimes also called a geometric progression.

Key terms:

• The first term is usually written as $u_1$.
• The common ratio is written as $r$.
• The $n$th term is written as $u_n$.

Geometric sequences are useful for modelling repeated percentage change or repeated scaling. Examples include population growth, depreciation, bacterial growth and compound interest.

Geometric sequences and series appear whenever there is repeated multiplication by the same factor.

• population growth by a fixed percentage each year
• depreciation of a car by a fixed percentage each year
• spread of disease
• compound interest
• repeated halving or doubling processes

In these contexts, the common ratio is often found from a percentage change.

• For example, a yearly increase of $5\%$ corresponds to $r=1.05$.
• A yearly decrease of $12\%$ corresponds to $r=0.88$.