Arithmetic Sequences
Learn arithmetic sequences for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-03). In an arithmetic progression each term is found by adding the same fixed amount, the common difference, to the term before it, so the terms grow or shrink in equal steps.
You will learn to test whether a list is arithmetic, find the common difference, use the nth term formula \(a + (n - 1)d\) to find any term or position, and see why the terms follow a straight-line pattern — handy for regular savings, wages and fixed price rises.
Theory
An arithmetic sequence (AP) changes by the same amount each step — the common difference \(d\). This Year 12 Mathematics Advanced topic (MAV-12-03) uses \(a_n=a+(n-1)d\) to find terms and solve problems, and recognises that the terms are a linear function of \(n\).
In an arithmetic sequence each term differs from the one before by a fixed common difference \(d=a_n-a_{n-1}\).
With first term \(a\), the \(n\)th term is \(a_n=a+(n-1)d\). Since this is linear in \(n\) (slope \(d\)), the terms plot as equally spaced points on a straight line. \(d>0\) increases; \(d<0\) decreases.
Method
- Find the common difference \(d\) (subtract consecutive terms) and the first term \(a\).
- Substitute into \(a_n=a+(n-1)d\).
- Solve for \(n\) if you are told the value of a term.
| \(d\) | \(=\) | \(4\) |
| \(a_{10}\) | \(=\) | \(3+9\times4=39\) |
| \(a_n\) | \(=\) | \(5+(n-1)(-3)=8-3n\) |
| \(a_{20}\) | \(=\) | \(-52\) |
| \(5n-1\) | \(=\) | \(99\) |
| \(n\) | \(=\) | \(20\) |
| \(d\) | \(=\) | \(2000\) |
| \(a_5\) | \(=\) | \(50000+4(2000)=58000\) |
| \(a_n\) | \(=\) | \(48000+2000n\) |
Common pitfalls
Frequently asked questions
What is the formula for the nth term of an arithmetic sequence?
It is a n equals a plus (n minus 1) times d, where a is the first term and d is the common difference.
How do you find the common difference?
Subtract any term from the term after it. If the sequence is 3, 7, 11 then d equals 7 minus 3, which is 4.
How do you find which term equals a given value?
Set the nth term formula equal to the value and solve for n. If n is a whole number, that position is the term.
Why is an arithmetic sequence linear?
Because a n equals a plus (n minus 1) d is a linear function of n with slope d, so the terms plot as equally spaced points on a straight line.