Arithmetic sequences | Year 11

01 / LEARN

The idea, step by step

Constant addition

An arithmetic sequence adds a fixed common difference $d$ at each step. Find it by subtracting consecutive terms in a consistent order. The difference may be positive, negative or zero.

If the first term is $a$ , reaching term $n$ takes $n - 1$ steps. The formula therefore contains $n - 1$ , not $n$ .

u_{n} = a + (n - 1) d

Unknown positions and two known terms

Subtract equations for two known terms. The difference between their indices is the number of equal steps between them. Once $d$ is found, substitute into either equation to find $a$ .

To check whether a specified value occurs, solve for $n$ and require a positive integer. A fractional solution is not a term position.

A common trap

The common difference is the second term divided by the first.

Instead: Use subtraction. Constant quotients identify geometric sequences.

02 / WORKED EXAMPLES

See the thinking

Example 1

Recovering a sequence

An arithmetic sequence has $u_{4} = 17, u_{10} = 47$ . Find its rule.

$6 d = 47 - 17 = 30$ , so $d = 5$ .
$a + 3 (5) = 17$ , so $a = 2$ and $u_{n} = 2 + 5 (n - 1)$ .

u_{n} = 5 n - 3

Example 2

Crossing a threshold

For $50, 47, 44, \dots$ , find the first negative term.

$u_{n} = 53 - 3 n$ .
$53 - 3 n < 0$ requires $n > 53 / 3$ ; the first allowed index is 18. Check $u_{17} = 2$ .

u_{18} = - 1

Check your understanding

If $a = 8, d = - 3$ , find $u_{6}$ .

Show answer

$- 7$ .

$8 + 5 (- 3) = - 7$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $d, u_{12}$ for $- 4, 3, 10, \dots$ .
2 marks
Find $a$ when $u_{8} = 31, d = 4$ .
2 marks
An arithmetic sequence has $u_{3} = 14, u_{9} = 38$ . Find $u_{20}$ .
2 marks
Is 104 a term of $5, 11, 17, \dots$ ?
2 marks
Insert three arithmetic means between 6 and 26.
2 marks
Row 1 needs 9 tiles; each later row needs 4 more. Which row first needs more than 100?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find $d, u_{12}$ for $- 4, 3, 10, \dots$ .
2 marks
$3 - (- 4) = 7$ ; $- 4 + 11 (7) = 73$ .
$d = 7, u_{12} = 73$ .
Find $a$ when $u_{8} = 31, d = 4$ .
2 marks
$a + 7 (4) = 31$ .
$3$ .
An arithmetic sequence has $u_{3} = 14, u_{9} = 38$ . Find $u_{20}$ .
2 marks
$6 d = 24$ gives $d = 4$ ; $a = 6$ ; $6 + 19 (4) = 82$ .
$82$ .
Is 104 a term of $5, 11, 17, \dots$ ?
2 marks
$u_{n} = 6 n - 1$ ; setting it to 104 gives $n = 17.5$ .
No.
Insert three arithmetic means between 6 and 26.
2 marks
Four equal intervals have size $(26 - 6) / 4 = 5$ .
$11, 16, 21$ .
Row 1 needs 9 tiles; each later row needs 4 more. Which row first needs more than 100?
3 marks
$9 + 4 (n - 1) > 100$ gives $n > 23.75$ .
Row 24.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics sequence and series content with introductory induction, regrouped into an original school-sized chapter.

Official curriculum source ↗

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