Numeric and geometric patterns (Grade 9) — Mathematics Grades 8–11

Spotting the pattern in a sequence, writing its general rule, and using the rule to find any term.

Gr 9 Term 1 — Numeric and geometric patternsTerm 1

Practise Numeric and geometric patterns in the app →

What gets asked

Extend a number pattern by finding the next few terms.
Find the general rule, the nth term, of a pattern.
Use a rule to find a term far along in the pattern.
Find the position of a given value in a pattern.

You must be able to

Find the constant difference or constant ratio in a simple pattern.
Write the general rule for a pattern in terms of n.
Substitute a position number into a rule to find that term.
Substitute a term value into a rule to find its position.

Traps that cost marks

Giving the constant difference alone as the general rule. The rule needs a starting value too. For $- 2; - 5; - 8; ...$ the rule is $- 3 n + 1$ , not just $- 3 n$ .
Swapping the term value and its position when working backwards. Put the given term value in for the answer in the rule, then solve for n, the position.
Counting the shapes instead of the objects the question actually asks about. Reread the question and count exactly what is asked for, such as matchsticks, not shapes.

Worked example

A pattern of tiles starts 5; 8; 11; 14; ... Find the general rule for the nth term.

The constant difference between terms is 3, so the rule starts with $3 n$ .
Test $n = 1$ : $3 (1) = 3$ , but the first term is 5, so add 2.
The general rule is $3 n + 2$ .

Extend a sequence

Spotting how a list of numbers grows, and finding a rule that reaches any place in it.

sequence: A list of numbers in a fixed order, such as $5; 8; 11; 14$ .
term: One of the numbers in a sequence.
position: Where a term sits in the sequence, counted from $1$ . It is written $n$ .
constant difference: The same amount you add on to reach every next term.
constant ratio: The same number you multiply by to reach every next term.
general rule: A rule that gives any term straight from its position.

A sequence is a list of numbers in a fixed order. Each number in it is a term, and every term has a position. The first term sits at position $1$ .

Test for a constant difference first. Subtract each term from the one after it. In $5; 8; 11$ both gaps come to $3$ , so you keep adding $3$ to go further.

If the gaps are not equal, test for a constant ratio instead. Divide each term by the one before it. In $2; 6; 18$ every term is $3$ times the last one, so you keep multiplying by $3$ .

Some sequences have neither. Then study the gaps themselves. In $2; 5; 10; 17$ the gaps run $3; 5; 7$ , and those gaps grow by $2$ each time.

Counting term by term is slow. A general rule gives any term straight from its position $n$ . With a constant difference of $d$ , the rule starts with $d n$ . Then add a fixed number so that position $1$ gives the first term.

Rules to remember

A rule must give the right term at every position, not only at the first

Examples

Write the next two terms:

3; 10; 17; 24; \dots

Worked answer

Find one gap: $10 - 3 = 7$ .
Check another gap: $24 - 17 = 7$ .
The difference is constant, so keep adding $7$ .
$24 + 7 = 31$ , and then $31 + 7 = 38$ .

Answer: $31$ and $38$

Two gaps were checked, so the pattern was not guessed from a single one.

Write the next term:

2; 5; 10; 17; \dots

Worked answer

The gaps are $3$ , $5$ and $7$ , so there is no constant difference.
There is no constant ratio either, since $5 \div 2$ and $10 \div 5$ differ.
But the gaps themselves grow by $2$ every time.
So the next gap must be $9$ .
The next term is $17 + 9 = 26$ .

Answer: $26$

The pattern lived in the gaps, not in the terms themselves.

A tile pattern uses

7; 11; 15; 19; \dots

tiles. Find the rule and the 20th term.

Worked answer

The constant difference is $4$ , so the rule starts with $4 n$ .
Test position $1$ : $4 \times 1 = 4$ , but the first term is $7$ .
It is $3$ short every time, so the general rule is $4 n + 3$ .
Check position $2$ : $4 \times 2 + 3 = 11$ .
For the 20th term put in $20$ : $4 \times 20 + 3 = 83$ .

Answer: $4 n + 3$ , and the 20th term is $83$

A tested rule reaches position $20$ at once, with no counting in between.

Traps

Giving the constant difference alone as the general rule. The rule needs a starting value too. For $- 2; - 5; - 8; ...$ the rule is $- 3 n + 1$ , not just $- 3 n$ .
Adding a gap to every sequence, so $2; 6; 18$ is carried on with $30$ . Test for a ratio too. Each term here is $3$ times the last, so the next one is $54$ .
Dropping the minus sign when the difference or the ratio is negative. Keep the sign. With ratio $- 0, 5$ , the term after $- 0, 25$ is $0, 125$ .
Using the first two terms only, and forcing a constant difference on the rest. Check every gap. In $2; 5; 10; 17$ the gaps are $3$ , $5$ and $7$ , all different.

Learn and practise “Extend a sequence” in the app →

More complex patterns

Working backwards to a position, patterns that grow faster, and rules read off a picture.

second difference: The gap between the gaps of a sequence.
square number: A number made by multiplying a whole number by itself, such as $1$ , $4$ , $9$ and $16$ .
geometric pattern: A pattern shown as pictures, where each picture uses more parts than the one before.

Sometimes you are given a term and asked for its position. Put that term value into the general rule, then solve for $n$ . The value goes where the answer goes, never where $n$ goes.

Undo the rule in reverse order. In $4 n + 3$ the $n$ was multiplied by $4$ , and then $3$ was added. So subtract the $3$ first, and divide by $4$ after that.

A position must come out as a whole number. If $n$ works out to a fraction, then the value you were handed is not in this sequence at all.

When the gaps are not constant, look at the second difference. If that is constant, the general rule uses $n^{2}$ . Squaring each position gives the square numbers $1; 4; 9; 16$ , and the rule adjusts them.

For a geometric pattern, count exactly what the question asks about. Turn the pictures into a table of position and count. Find the rule from the table, then test it on a later picture.

Rules to remember

A constant second difference means the rule uses $n^{2}$

Examples

The rule for a sequence is

4 n + 3

. In what position is the term

83

Worked answer

Put the term value in where the answer goes: $4 n + 3 = 83$ .
Subtract $3$ from both sides: $4 n = 80$ .
Divide both sides by $4$ : $n = 20$ .
So $83$ is the term at position $20$ .

Answer: $n = 20$

The $83$ was a term value, so it went in on the answer side of the rule.

Find the general rule for

3; 6; 11; 18; \dots

Worked answer

The gaps are $3$ , $5$ and $7$ , so they are not constant.
Those gaps grow by $2$ each time, so the second difference is $2$ .
A second difference of $2$ means the rule uses $n^{2}$ .
The square numbers are $1; 4; 9; 16$ , and each is $2$ short.
So the general rule is $n^{2} + 2$ .

Answer: $n^{2} + 2$

The second difference pointed at $n^{2}$ , and one comparison fixed the rest.

Squares are joined in a row with matchsticks. Picture

1

uses

4

, picture

2

uses

7

, picture

3

uses

10

. How many for picture

10

Worked answer

Count matchsticks, not squares.
Make a table: position $1$ gives $4$ , position $2$ gives $7$ .
The constant difference is $3$ , so the rule starts with $3 n$ .
Test position $1$ : $3 \times 1 + 1 = 4$ , so the rule is $3 n + 1$ .
For picture $10$ : $3 \times 10 + 1 = 31$ .

Answer: $31$ matchsticks

The table turned pictures into numbers, and the rule was tested before it was used.

Traps

Swapping the term value and its position when working backwards. Put the given term value in for the answer in the rule, then solve for n, the position.
Dividing before subtracting when undoing a rule. Undo in reverse order. For $4 n + 3$ , subtract the $3$ first and divide by $4$ after.
Deciding a sequence has no rule because its gaps are not constant. Work out the second difference. If that one is constant, the rule uses $n^{2}$ .
Counting the shapes instead of the objects the question actually asks about. Reread the question and count exactly what is asked for, such as matchsticks, not shapes.

Learn and practise “More complex patterns” in the app →

Grade 9 Numeric and geometric patterns

What gets asked

You must be able to

Traps that cost marks

Worked example

Extend a sequence

Rules to remember

Examples

Traps

More complex patterns

Rules to remember

Examples

Traps

About this material