Arithmetic sequences

Arithmetic sequences (APs): the common difference, testing for an AP, the formula T(n) = a + (n − 1)d, counting the terms of an AP, and the link with linear functions.

Common difference

An arithmetic sequence (or arithmetic progression, AP) adds the same number each time. That number is the common difference $d$.

For example, $17, 12, 7, 2, \ldots$ is an AP with $d = -5$. But $1, 4, 9, 16, \ldots$ is not, because the differences are $3, 5, 7, \ldots$

The nth term

nth term of an AP

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

An AP lies on a straight line

Plot each term T_n against n. The points lie on a line with gradient d, which crosses n = 1 at height a.

Try this

Make $d$ negative. What happens to the terms?

Example 1●

Find $T_{25}$ for the AP $17, 12, 7, \ldots$

$a = 17$ and $d = -5$.

$T_{25} = 17 + 24 \times (-5) = -103$

How many terms?

Example 2●●

How many terms are in the AP $6, 13, 20, \ldots, 342$?

$a = 6$, $d = 7$. Solve $T_n = 342$.

$6 + 7(n - 1) = 342$, so $7(n - 1) = 336$ and $n - 1 = 48$.

$n = 49$ terms.

Example 3: the first negative term●●

Find the first negative term of $85, 81, 77, \ldots$

$T_n = 85 - 4(n - 1) = 89 - 4n$.

$89 - 4n < 0$ when $n > 22.25$, so the first negative term is $T_{23}$.

$T_{23} = 89 - 92 = -3$.

Problems with two pieces of information

Example 4●●●

An AP has $T_5 = 23$ and $T_{12} = 51$. Find $a$ and $d$.

$a + 4d = 23$ and $a + 11d = 51$.

$7d = 28$, so $d = 4$ and then $a = 23 - 16 = 7$.

Subtract the first equation from the second.

Flashcards

Common difference

$d = T_{n+1} - T_n$

$n$th term of an AP

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

Graph of an AP

Points on a straight line with gradient $d$.

Practice

Find $T_{25}$ for the AP $17, 12, 7, \ldots$
How many terms are in $6, 13, 20, \ldots, 342$?
Find the first negative term of $85, 81, 77, \ldots$
An AP has $T_5 = 23$ and $T_{12} = 51$. Find $d$.
Which sequence is arithmetic?
The $n$th term of the AP $2, 5, 8, \ldots$ is:
Find $x$ if $x - 3$, $x + 1$ and $2x - 2$ are consecutive terms of an AP.
Plotted against $n$, the terms of an AP lie on:
How many terms of $1, 3, 5, \ldots$ are less than $200$?