Arithmetic sequences (the general term)

Covers AS91258

What an arithmetic sequence is

An arithmetic sequence goes up (or down) by the same amount every step.
That fixed step is the common difference, $d$ — found by subtracting any term from the one after it.
The starting value is the first term, $a$ (sometimes written $u_1$ ).
Example — $3, 7, 11, 15, \ldots$ has $a = 3$ and $d = 4$ .

A bar chart of the sequence 3, 7, 11, 15 where each bar is 4 taller than the last, and the tops lie on a straight line.

The general term

The general term (or $n$ -th term) is a formula for any term without listing them all:

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

$u_n$ — the term in position $n$ .
$a$ — the first term.
$d$ — the common difference.
$n$ — the position you want ( $n = 1$ is the first term).
The " $(n - 1)$ " is there because you take $d$ steps starting after the first term — to reach the 5th term you add $d$ four times, not five.

Finding a term

Identify $a$ and $d$ from the sequence.
Substitute the position $n$ into $u_n = a + (n - 1)d$ .

Worked ExampleFinding a distant term

A theatre has 24 seats in the front row and 3 more in each row behind it. How many seats are in the 12th row?

Step 1 — Identify $a$ and $d$

The rows form an arithmetic sequence with first term $a = 24$ and common difference $d = 3$ .

Step 2 — Substitute $n = 12$

$u_{12} = 24 + (12 - 1)(3) = 24 + 33 = 57$

So the 12th row has 57 seats.

Working backwards to find $a$ , $d$ or $n$

If you know two terms, set up equations with $u_n = a + (n-1)d$ and solve.
To find which term equals a given value, substitute the value for $u_n$ and solve for $n$ .

Worked ExampleFinding the common difference

In an arithmetic sequence the 3rd term is 17 and the 7th term is 41. Find $a$ and $d$ .

Step 1 — Write an equation for each term

$u_3 = a + 2d = 17 \qquad u_7 = a + 6d = 41$

Step 2 — Subtract to eliminate $a$

$(a + 6d) - (a + 2d) = 41 - 17 \;\Rightarrow\; 4d = 24 \;\Rightarrow\; d = 6$

Step 3 — Back-substitute for $a$

$a + 2(6) = 17 \;\Rightarrow\; a = 5$

The sequence starts at $5$ and goes up by $6$ .

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

An arithmetic sequence has first term $a = 8$ and common difference $d = 5$ . Find the 10th term.

Merit

The 4th term of an arithmetic sequence is 20 and the 9th term is 45. Find the first term and common difference.

Excellence

A runner increases her weekly distance by the same amount each week. In week 2 she runs 18 km and in week 6 she runs 30 km. In which week does she first run at least 45 km?

Arithmetic sequences (the general term)

Covers AS91258

What an arithmetic sequence is

An arithmetic sequence goes up (or down) by the same amount every step.
That fixed step is the common difference, $d$ — found by subtracting any term from the one after it.
The starting value is the first term, $a$ (sometimes written $u_1$ ).
Example — $3, 7, 11, 15, \ldots$ has $a = 3$ and $d = 4$ .

A bar chart of the sequence 3, 7, 11, 15 where each bar is 4 taller than the last, and the tops lie on a straight line.

The general term

The general term (or $n$ -th term) is a formula for any term without listing them all:

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

$u_n$ — the term in position $n$ .
$a$ — the first term.
$d$ — the common difference.
$n$ — the position you want ( $n = 1$ is the first term).
The " $(n - 1)$ " is there because you take $d$ steps starting after the first term — to reach the 5th term you add $d$ four times, not five.

Finding a term

Identify $a$ and $d$ from the sequence.
Substitute the position $n$ into $u_n = a + (n - 1)d$ .

Worked ExampleFinding a distant term

A theatre has 24 seats in the front row and 3 more in each row behind it. How many seats are in the 12th row?

Step 1 — Identify $a$ and $d$

The rows form an arithmetic sequence with first term $a = 24$ and common difference $d = 3$ .

Step 2 — Substitute $n = 12$

$u_{12} = 24 + (12 - 1)(3) = 24 + 33 = 57$

So the 12th row has 57 seats.

Working backwards to find $a$ , $d$ or $n$

If you know two terms, set up equations with $u_n = a + (n-1)d$ and solve.
To find which term equals a given value, substitute the value for $u_n$ and solve for $n$ .

Worked ExampleFinding the common difference

In an arithmetic sequence the 3rd term is 17 and the 7th term is 41. Find $a$ and $d$ .

Step 1 — Write an equation for each term

$u_3 = a + 2d = 17 \qquad u_7 = a + 6d = 41$

Step 2 — Subtract to eliminate $a$

$(a + 6d) - (a + 2d) = 41 - 17 \;\Rightarrow\; 4d = 24 \;\Rightarrow\; d = 6$

Step 3 — Back-substitute for $a$

$a + 2(6) = 17 \;\Rightarrow\; a = 5$

The sequence starts at $5$ and goes up by $6$ .

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

An arithmetic sequence has first term $a = 8$ and common difference $d = 5$ . Find the 10th term.

Merit

The 4th term of an arithmetic sequence is 20 and the 9th term is 45. Find the first term and common difference.

Excellence

A runner increases her weekly distance by the same amount each week. In week 2 she runs 18 km and in week 6 she runs 30 km. In which week does she first run at least 45 km?

Arithmetic sequences (the general term)

What an arithmetic sequence is

The general term

Finding a term

Step 1 — Identify aaa and ddd

Step 2 — Substitute n=12n = 12n=12

Working backwards to find aaa, ddd or nnn

Step 1 — Write an equation for each term

Step 2 — Subtract to eliminate aaa

Step 3 — Back-substitute for aaa

Test yourself

Practice by grade

Arithmetic sequences (the general term)

What an arithmetic sequence is

The general term

Finding a term

Step 1 — Identify aaa and ddd

Step 2 — Substitute n=12n = 12n=12

Working backwards to find aaa, ddd or nnn

Step 1 — Write an equation for each term

Step 2 — Subtract to eliminate aaa

Step 3 — Back-substitute for aaa

Test yourself

Practice by grade

Step 1 — Identify $a$ and $d$

Step 2 — Substitute $n = 12$

Working backwards to find $a$ , $d$ or $n$

Step 2 — Subtract to eliminate $a$

Step 3 — Back-substitute for $a$

Step 1 — Identify $a$ and $d$

Step 2 — Substitute $n = 12$

Working backwards to find $a$ , $d$ or $n$

Step 2 — Subtract to eliminate $a$

Step 3 — Back-substitute for $a$