Year 8 Maths Graphing Linear Equations

Finding Rules for Number Patterns

20 practice questions 1 video lesson Theory + worked examples

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

Watch 1 video(s)

Finding Rules for Number Patterns - Video - Making an algebraic rule from a simple pattern Watch

Theory

A pattern that goes up or down by the same step has a rule: multiply the term number by the step, then add or take away a number so the rule fits. For $10, 7, 4, 1, \ldots$ the step is $-3$ and the rule is $T_n = -3n + 13$, so $T_{10} = -17$. This Year 8 maths page shows how.

Find the step. Take each term from the one after it. The same answer every time means the pattern has a constant step. A pattern that goes down has a negative step.

Describe it in words with the start number and the step: “start at $20$ and take away $3$ each time”.

Build the rule. With step $s$, the rule starts $T = sn$, where $n$ is the term number. Put $n = 1$ into $sn$ and compare with the 1st term; the difference is the number to add. For $20, 17, 14, \ldots$: $-3 \times 1 = -3$ but $T_1 = 20$, so add $23$: $T = -3n + 23$.

From a table, $y = (\text{step}) \times x + (\text{number})$. When the table starts at $x = 0$, the number to add is the first $y$-value.

Use the rule. Substitute a term number to reach a far term. To go the other way, make the rule equal the value and solve. A term number must be a whole number.

Not every pattern has a constant step. In $2, 6, 18, 54, \ldots$ each term is multiplied by $3$; in $3, 4, 7, 11, 18, \ldots$ each term is the sum of the two before it. Look at how each term is made; if a rule is given, substitute into it.

Each border starts with $3$ grey tiles and each section adds $2$ blue tiles: $T = 2n + 3$, giving $5, 7, 9$.

$4, 11, 18, 25, \ldots$: start at $4$ and add $7$ each time.

Building the rule for the nth term:

Pattern	Working	Rule
$5, 7, 9, \ldots$	step $2$; $2 \times 1 = 2$, add $3$	$T = 2n + 3$
$10, 7, 4, \ldots$	step $-3$; $-3 \times 1 = -3$, add $13$	$T = -3n + 13$

          Check the step between every pair. If the step changes, the pattern has no rule of this kind.
        

          Check the rule in another term. For \(T = -3n + 13\): \(T_4 = -12 + 13 = 1\), which matches.
        

Finding the rule for the nth term

Find the step $s$ and check it is the same between every pair.
Write $T = sn$ and work out $s \times 1$.
Compare with the 1st term: add (or take away) the difference.
Check the rule in another term.

Example 1 — Missing terms

Find the pronumerals in the pattern $1, 3, 5, 7, a, b$.

Solution

$a$	$=$	$7 + 2 = 9$
$b$	$=$	$9 + 2 = 11$

a = 9, b = 11

The terms go up by $2$: $a = 9$ and $b = 11$.

Example 2 — The nth term of a falling pattern

Find a rule for the $n$th term $T_n$ of $10, 7, 4, 1, \ldots$, then find $T_{10}$.

Solution

The terms go down by $3$, so $T_n = -3n + b$. With $n = 1$: $10 = -3 + b$, so $b = 13$.

$T_n$	$=$	$-3n + 13$
$T_{10}$	$=$	$-3 \times 10 + 13 = -17$

T = - 3 n + 13 - 17

$T_n = -3n + 13$ and $T_{10} = -17$.

Example 3 — Continue and describe

For the pattern $4, 11, 18, 25, \ldots$
(i) What is the step?
(ii) Write the next two terms.
(iii) Describe the pattern in words.

Solution

$11 - 4 = 7$, $18 - 11 = 7$, $25 - 18 = 7$.

(i) $7$
(ii) $32$ and $39$
(iii) Start at $4$ and add $7$ each time.

Example 4 — A falling table

Find the rule linking $x$ and $y$, then find $y$ when $x = 12$.

$x$	$0$	$1$	$2$	$3$	$4$
$y$	$9$	$5$	$1$	$-3$	$-7$

Solution

The step is $5 - 9 = -4$. The table starts at $x = 0$, so the number to add is $9$.

$y$	$=$	$-4x + 9$
$-4 \times 12 + 9$	$=$	$-39$

y = - 4 x + 9 - 39

$y = -4x + 9$, and $y = -39$ when $x = 12$. Check $x = 4$: $-16 + 9 = -7$.

Example 5 — The nth term

For the pattern $7, 12, 17, 22, \ldots$
(i) Find a rule for $T$.
(ii) Find the $20$th term.
(iii) Is $100$ a term?

Solution

The step is $5$. At $n = 1$, $5 \times 1 = 5$ but $T = 7$, so add $2$.

$T$	$=$	$5n + 2$
$5 \times 20 + 2$	$=$	$102$
$5n + 2$	$=$	$100$
$n$	$=$	$19.6$

T = 5 n + 2102 no

(i) $T = 5n + 2$
(ii) $102$
(iii) No: $n = 19.6$ is not a whole number.

Example 6 — A hire charge

A kayak shop charges a deposit plus the same amount each hour: $1$ h $\$43$, $2$ h $\$61$, $3$ h $\$79$, $4$ h $\$97$.
(i) Find a rule for the cost $C$.
(ii) Find the cost of $6$ hours.
(iii) A hire cost $\$205$. How many hours?

Solution

Each hour adds $61 - 43 = 18$. At $h = 1$, $18 \times 1 = 18$ but $C = 43$, so add $25$ (the deposit).

$C$	$=$	$18h + 25$
$18 \times 6 + 25$	$=$	$133$
$18h$	$=$	$205 - 25 = 180$
$h$	$=$	$10$

C = 18 h + 25 $ 13310 hours

(i) $C = 18h + 25$
(ii) $\$133$
(iii) $10$ hours

Example 7 — No constant step

Find the next two terms.
(i) $2, 6, 18, 54, \ldots$
(ii) $3, 4, 7, 11, 18, \ldots$

Solution

(i) Each term is $3$ times the one before: $162$ and $486$.
(ii) Each term is the sum of the two before it: $11 + 18 = 29$, then $18 + 29 = 47$.

Example 8 — When two patterns meet

Pattern A is $4, 7, 10, \ldots$ with rule $T = 3n + 1$. Pattern B is $34, 31, 28, \ldots$
(i) Find a rule for pattern B.
(ii) For which $n$ are the terms equal, and what is that term?

Solution

B goes down by $3$: at $n = 1$, $-3 \times 1 = -3$ but $T = 34$, so add $37$.

$T$	$=$	$37 - 3n$
$3n + 1$	$=$	$37 - 3n$
$6n$	$=$	$36$
$n$	$=$	$6$

T = 37 - 3 n n = 619

(i) $T = 37 - 3n$
(ii) $n = 6$, where both terms are $3 \times 6 + 1 = 19$.

Common pitfalls

Using the first term as the number to add. For $7, 12, 17, \ldots$ the rule is $5n + 2$, not $5n + 7$. Compare $5 \times 1$ with $7$.

Forgetting the negative step. A falling pattern such as $10, 7, 4$ has step $-3$.

Accepting a decimal term number. If $n = 19.6$, the value is not in the pattern.

Using the method on a pattern with no constant step. $2, 6, 18, 54$ is multiplied by $3$ each time.

Frequently asked questions

How do I find the rule for a number pattern?

Find the step, write T = step × n, then add or subtract the number that makes the 1st term correct. Check with another term.

What does the nth term mean?

The term in position n. T₁ is the first term, T₁₀ the tenth.

How do I check whether a number is in a pattern?

Set the rule equal to the number and solve for n. It is a term only if n is a positive whole number.

What if the pattern goes down?

The step is negative, so the rule starts with a negative multiple of n, such as T = −3n + 13.

Do all patterns have a rule like T = 5n + 2?

No. Only patterns with a constant step. Others, such as 2, 6, 18, 54, are made in different ways.

Pattern	Working	Rule
\(5, 7, 9, \ldots\)	step \(2\); \(2 \times 1 = 2\), add \(3\)	\(T = 2n + 3\)
\(10, 7, 4, \ldots\)	step \(-3\); \(-3 \times 1 = -3\), add \(13\)	\(T = -3n + 13\)

\(a\)	\(=\)	\(7 + 2 = 9\)
\(b\)	\(=\)	\(9 + 2 = 11\)

\(T_n\)	\(=\)	\(-3n + 13\)
\(T_{10}\)	\(=\)	\(-3 \times 10 + 13 = -17\)

\(y\)	\(=\)	\(-4x + 9\)
\(-4 \times 12 + 9\)	\(=\)	\(-39\)

\(T\)	\(=\)	\(5n + 2\)
\(5 \times 20 + 2\)	\(=\)	\(102\)
\(5n + 2\)	\(=\)	\(100\)
\(n\)	\(=\)	\(19.6\)

\(C\)	\(=\)	\(18h + 25\)
\(18 \times 6 + 25\)	\(=\)	\(133\)
\(18h\)	\(=\)	\(205 - 25 = 180\)
\(h\)	\(=\)	\(10\)

\(T\)	\(=\)	\(37 - 3n\)
\(3n + 1\)	\(=\)	\(37 - 3n\)
\(6n\)	\(=\)	\(36\)
\(n\)	\(=\)	\(6\)

Finding Rules for Number Patterns

📖 Theory

Finding the rule for the nth term

Common pitfalls

Frequently asked questions

Theory