Arithmetic sequences | Year 11

01 / LEARN

Understand the idea.

Constant addition

An arithmetic sequence changes by the same amount each time. Calculate the common difference as later term minus earlier term. A negative difference makes the terms decrease; a zero difference gives a constant sequence.

From $u_{1} = a$ to $u_{n}$ there are $n - 1$ equal steps, giving $u_{n} = a + (n - 1) d$ . The discrete points lie on a straight line, but only permitted integer indices are sequence positions.

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

Five unjoined gold points plot sequence terms four, seven, ten, thirteen and sixteen against indices one through five. Every term is labelled; the horizontal axis is index n and the vertical axis is term.

Unknown positions and two known terms

Two known terms determine the common difference: $u_{q} - u_{p} = (q - p) d$ . The index gap counts the number of steps. Divide the change in value by that gap, then substitute into either term equation to recover $a$ .

If a question asks when a value is reached or crossed, solve the resulting equation or inequality for $n$ . Require a positive integer, and check the neighbouring indices when identifying the first term satisfying an inequality.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Recovering a sequence

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

Worked solution

1
Write equations for the supplied terms.
$\begin{aligned} a + 3 d & = 17 \\ a + 9 d & = 47 \end{aligned}$
Term four is three steps after the first; term ten is nine steps after it.
2
Subtract to isolate the difference.
$\begin{aligned} 6 d & = 47 - 17 = 30 \\ d & = 5 \end{aligned}$
There are six equal steps between indices four and ten.
3
Recover the first term.
$a = 17 - 3 (5) = 2$
Use the fourth-term equation with the known difference.
4
Form and simplify the rule.
$u_{n} = 2 + 5 (n - 1) = 5 n - 3$
The explicit rule starts at two and adds five per step.

Answer

$u_{n} = 5 n - 3$ .

EXAMPLE 02

Crossing a threshold

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

Worked solution

1
Write the decreasing term rule.
$u_{n} = 50 - 3 (n - 1) = 53 - 3 n$
The common difference is negative three.
2
Require a negative value.
$\begin{aligned} 53 - 3 n & < 0 \\ n & > \frac{53}{3} \end{aligned}$
Dividing by a negative number reverses the inequality.
3
Choose and check the first allowed index.
$\begin{aligned} u_{17} & = 53 - 51 = 2 \\ u_{18} & = 53 - 54 = - 1 \end{aligned}$
The terms decrease, and the sign first changes between these consecutive indices.

Answer

$u_{18} = - 1$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$- 7$ .

Term six lies five steps after term one, so $u_{6} = 8 + 5 (- 3) = - 7$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 $\frac{26 - 6}{4} = 5$ .
$11, 16, 21$ .
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.

Checking saved progress…

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 ↗

In this lesson

Understand the idea.

Constant addition

Unknown positions and two known terms

See the method in action.

Recovering a sequence

Write equations for the supplied terms.

Subtract to isolate the difference.

Recover the first term.

Form and simplify the rule.

Crossing a threshold

Write the decreasing term rule.

Require a negative value.

Choose and check the first allowed index.

Try it yourself.

Check your understanding

Keep practising