Geometric Sequences
Learn geometric sequences for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-03). In a geometric progression each term is found by multiplying the previous one by the same number, the common ratio, so the terms grow or shrink by a constant factor.
You will learn to test whether a list is geometric, find the common ratio, use the nth term formula \(ar^{n-1}\) to find a term or its position, and recognise the exponential pattern behind it — the maths of doubling, halving and percentage change.
Theory
A geometric sequence (GP) multiplies by the same amount each step — the common ratio \(r\). This Year 12 Mathematics Advanced topic (MAV-12-03) uses \(a_n=ar^{\,n-1}\) to find terms and solve growth and decay problems.
In a geometric sequence each term is a fixed multiple of the one before: the common ratio is \(r=\dfrac{a_n}{a_{n-1}}\).
With first term \(a\), the \(n\)th term is \(a_n=ar^{\,n-1}\). \(|r|>1\) grows; \(0<|r|<1\) decays toward \(0\); a negative \(r\) makes the terms alternate in sign.
Method
- Find the common ratio \(r\) (divide a term by the previous one) and the first term \(a\).
- Substitute into \(a_n=ar^{\,n-1}\).
- Apply in context: a repeated percentage or fraction of the previous value.
| \(r\) | \(=\) | \(3\) |
| \(a_6\) | \(=\) | \(2\times3^5=486\) |
| \(a_n\) | \(=\) | \(16\left(\tfrac12\right)^{n-1}\) |
| \(a_5\) | \(=\) | \(16\times\tfrac{1}{16}=1\) |
| \(h\) | \(=\) | \(10\times0.8^3\) |
| \(=\) | \(5.12\text{ m}\) |
| \(r\) | \(=\) | \(2\) |
| \(a_7\) | \(=\) | \(3\times2^6=192\) |
| \(a_n\) | \(=\) | \(3\times2^{\,n-1}\) |
Common pitfalls
Frequently asked questions
What is the formula for the nth term of a geometric sequence?
It is a n equals a times r to the power n minus 1, where a is the first term and r is the common ratio.
How do you find the common ratio?
Divide any term by the term before it. For 2, 6, 18 the ratio is 6 divided by 2, which is 3.
What is the difference between an arithmetic and geometric sequence?
An arithmetic sequence adds a fixed amount each step, while a geometric sequence multiplies by a fixed ratio each step.
Can the common ratio be a fraction?
Yes. A ratio between 0 and 1 makes a decaying sequence, such as 16, 8, 4 with ratio one half.