How Do You Find the Nth Term of an Arithmetic Sequence?

Use the nth-term formula $u_n=u_1+(n-1)d$, where $u_1$ is the first term and $d$ is the common difference. Identify these two values, substitute the required term number for $n$, and calculate.

This is SL 1.2, Sequences and Sigma Notation, studied by both SL and HL Mathematics: Applications and Interpretation students. It can be tested in either Paper 1 or Paper 2, and a GDC is permitted on every Mathematics AI paper.

How the Formula Works

An arithmetic sequence changes by the same amount between consecutive terms. This fixed amount is the common difference:

$$d=u_2-u_1$$

The factor $n-1$ is essential because reaching the first term requires zero steps, the second term requires one step, and the nth term requires $n-1$ steps.

Worked Example

Consider the sequence

$$7,\ 11,\ 15,\ 19,\ldots$$

The first term is $u_1=7$, and the common difference is

$$d=11-7=4.$$

Therefore,

$$u_n=7+4(n-1)=4n+3.$$

To find the 25th term, substitute $n=25$:

$$u_{25}=7+4(25-1)=103.$$

If the common difference is not stated, subtract consecutive terms. For example, if $u_4=18$ and $u_9=38$, there are $9-$ equal steps, so . Then , giving .

You may also be asked whether a value belongs to the sequence. Set the nth-term expression equal to that value and solve for $n$. It is a term only if $n$ is a positive integer. For example, $4n+2=70$ gives $n=17$, so 70 is the 17th term.

Exam Technique

For the IB command term find, show the relevant stages: identify $u_1$ and $d$, write the formula, and substitute correctly. If technology is used, the syllabus specifically expects you to identify the first term and common difference.

A common misconception is using $u_n=u_1+nd$. This is an off-by-one error: always use $n-1$, because no common difference is added to reach the first term.