Arithmetic progressions: the nth term

6 minAS Level1.6.21.6.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can recognise an arithmetic progression, find any term with $u_n = a + (n - 1)d$ , and use $2b = a + c$ to test whether three numbers are in arithmetic progression.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$n$	the position of a term: 1st, 2nd, 3rd, …
$u_n$	the $n$ th term, e.g. $u_4$ is the fourth term
$u_{n+1}$	the term straight after $u_n$
$a$	the first term, $u_1$
$d$	the common difference: the amount added each time

Same step, every step

A staircase where every step is the same height.
A taxi meter that adds the same fare every kilometre.
A savings jar that gets $5 every week.

That's an arithmetic progression (AP): you get each term by adding the same number to the one before.

Cambridge calls the first term $a$ and the fixed step $d$ . Example: $3, 7, 11, 15, \dots$ has $a = 3$ and $d = 4$ .

Definition

Arithmetic progression

An arithmetic progression is a sequence in which each term after the first is obtained by adding a constant, the common difference $d$ , to the previous term:

$u_{n+1} - u_n = d \quad \text{for every } n.$

$d$ can be positive, negative or a fraction. For example $20, 17, 14, 11, \dots$ is an AP with $a = 20$ and $d = -3$ .

The nth term

The $n$ th term is the first term plus $(n - 1)$ steps:

$u_n = a + (n - 1)d$

See it with numbers. In $3, 7, 11, 15, \dots$ (so $a = 3$ , $d = 4$ ):

$u_1 = 3$ (no steps yet)
$u_2 = 3 + 4$ (one step)
$u_3 = 3 + 4 + 4 = 3 + 2 \times 4$ (two steps)
$u_4 = 3 + 3 \times 4 = 15$ (three steps)

The number of steps is always one less than the position. So $u_n = a + (n - 1)d$ .

Same reason a 10 m fence with posts every metre needs 11 posts: the first post is already there before you take any steps.

To recognise an AP, check that the difference between consecutive terms is the same all the way along, not just for the first pair.

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.