Number patterns (Grade 10) — Mathematics Grades 8–11

Find and use the general term of a number pattern with a constant difference.

Gr 10 Term 1 - Number patternsTerm 1 - 1 week (CAPS)

Practise Number patterns in the app →

What gets asked

Find the constant difference between consecutive terms of a pattern.
Work out the general term (the rule) for a pattern with a constant difference.
Use the general term to find a value far along the pattern.
Decide whether a given number is actually a term of the pattern.

You must be able to

Subtract consecutive terms to check that the difference stays constant.
Build the general term from the first term and the constant difference.
Substitute a position number into the general term to find that value.
Test whether a given number fits the general term at a whole-number position.

Traps that cost marks

Using the constant difference as the fixed number in the rule, instead of as the multiplier of n. The difference multiplies $n$ . Find the fixed number separately by checking $n = 1$ .
Writing the rule as just the difference times n, without checking it gives the first term. Test your rule at $n = 1$ . Adjust the constant until it matches the first term.
Accepting a fraction as the answer when asked which position a value sits at. A position must be a whole number. A fraction means that value is not in the pattern.

Worked example

A pattern begins 5 ; 8 ; 11 ; 14 ; ... Find the general term.

The constant difference is $8 - 5 = 3$ .
Try $T_{n} = 3 n$ . At $n = 1$ this gives $3$ , but the first term is $5$ , so add $2$ .
General term: $T_{n} = 3 n + 2$ .

Constant difference patterns

Patterns that grow by the same amount each time, and the one rule that describes all of them.

term: One of the numbers in a pattern. The first one is written $T_{1}$ .
position: Where a term sits in the list. The first term is at position $1$ .
constant difference: The amount added each time to reach the next term. It never changes.
general term: A rule that gives any term from its position. We write it $T_{n}$ .

A number pattern is a list written in order. Each number in it is a term, and every term has a position. To see how the list grows, subtract each term from the one after it. Always take later minus earlier.

If all those subtractions give the same answer, the pattern has a constant difference. Check every pair, not only the first. A falling pattern gives a negative difference, which is fine.

Where the difference is constant, the general term is linear. It looks like a number times $n$ , plus a fixed number. The constant difference is the number that multiplies $n$ . It is not the fixed part.

Find the fixed part by testing position $1$ . See what the multiplying part gives there. Compare it to the real first term, then close the gap.

Not every rising pattern works this way. In $2$ ; $5$ ; $10$ ; $17$ the gaps are $3$ , $5$ and $7$ . Those gaps grow, so there is no constant difference here.

Rules to remember

$T_{n} = d n + c$ , where $d$ is the constant difference
$d = T_{2} - T_{1}$

Examples

Find the constant difference in

20

;

17

;

14

;

11

Worked answer

Take the later term minus the earlier one: $17 - 20 = - 3$ .
Check the next pair: $14 - 17 = - 3$ .
And the last pair: $11 - 14 = - 3$ .
Every gap is the same, so the difference is constant.

Answer: $- 3$

Subtracting the other way round would give $3$ , which points the wrong way.

Find the general term of

5

;

9

;

13

;

17

Worked answer

The difference is $9 - 5 = 4$ , and it holds all the way.
So the rule starts with $4 n$ , since the difference multiplies $n$ .
At position $1$ that gives $4$ , but the real first term is $5$ .
Add $1$ to close the gap: $T_{n} = 4 n + 1$ .
Test it at $n = 3$ : $4 (3) + 1 = 13$ , which matches.

Answer: $T_{n} = 4 n + 1$

The $4$ went in front of $n$ , and position $1$ gave the $1$ .

Find the general term of

30

;

26

;

22

;

18

Worked answer

The difference is $26 - 30 = - 4$ , so the rule starts with $- 4 n$ .
At position $1$ that gives $- 4$ , but the real first term is $30$ .
From $- 4$ up to $30$ you add $34$ .
So the general term is $T_{n} = - 4 n + 34$ .
Test it at $n = 4$ : $- 4 (4) + 34 = 18$ .

Answer: $T_{n} = - 4 n + 34$

A falling pattern gives a negative difference, so the fixed part beats the first term.

Traps

Using the constant difference as the fixed number in the rule, instead of as the multiplier of n. The difference multiplies $n$ . Find the fixed number separately by checking $n = 1$ .
Writing the rule as just the difference times n, without checking it gives the first term. Test your rule at $n = 1$ . Adjust the constant until it matches the first term.
Calling the difference in $20$ ; $17$ ; $14$ a positive $3$ . Take the later term minus the earlier one: $17 - 20 = - 3$ .
Checking only the first pair of $2$ ; $5$ ; $10$ ; $17$ and calling it constant. Check every pair. Here $5 - 2 = 3$ but $10 - 5 = 5$ .

Learn and practise “Constant difference patterns” in the app →

Terms and positions

Jumping straight to the hundredth item without writing the ninety-nine before it.

substitute: To put a number in place of a letter, then work the answer out.
cycle: The block of items that comes round again and again.
remainder: What is left over after dividing as many whole times as you can.

Once you have the general term, you never count along again. Substitute the position number into the rule and the term drops out. Put in the position, never the value of a term.

Some questions run the other way. You are given a value and asked where it sits. Set the rule equal to it and solve. If the position comes out a fraction, that value is not in the pattern.

Patterns also arrive as diagrams or stories. Count the items in the first figure, then in the second. The increase from one figure to the next is the constant difference, not the total.

Repeating patterns work differently. Nothing grows; a block just comes round again. Count the items in that cycle, then divide the position by it. The remainder says which item you land on.

Beads run red, green, blue, yellow, over and over: a cycle of $4$ . Divide bead $30$ by $4$ and the remainder is $2$ . So bead $30$ is green, the second colour. A remainder of $0$ means the last item.

Rules to remember

$T_{50}$ means the term at position $50$
Put a position in to get a term; set the rule equal to get a position

Examples

The general term of a pattern is

T_{n} = 7 n - 2

. Find the

50

th term.

Worked answer

The position is $50$ , so substitute $50$ in place of $n$ .
$T_{50} = 7 (50) - 2$ .
$7 (50) = 350$ , then $350 - 2 = 348$ .

Answer: $348$

Counting $50$ terms by hand invites a slip; the rule lands there in one go.

A pattern reads

6

;

11

;

16

;

21

. Is

96

a term, and where?

Worked answer

The difference is $11 - 6 = 5$ , so the rule starts with $5 n$ .
At position $1$ it gives $5$ , one short, so $T_{n} = 5 n + 1$ .
Set the rule equal to the value: $5 n + 1 = 96$ .
Then $5 n = 95$ , so $n = 19$ .
That is a whole number, so $96$ sits at position $19$ .

Answer: yes, at position $19$

A whole-number position proves the value really is in the pattern.

One school desk seats

4

learners. Each extra desk in the row seats

2

more. How many desks seat

30

learners?

Worked answer

Seats go $4$ , then $6$ , then $8$ , so the difference is $2$ .
The rule starts with $2 n$ , which gives $2$ at position $1$ , not $4$ .
Add $2$ , so the rule is $T_{n} = 2 n + 2$ .
Now solve $2 n + 2 = 30$ , so $2 n = 28$ and $n = 14$ .
Desks were asked for, and $14$ is a desk count.

Answer: $14$ desks

The difference came from the seats each extra desk adds, not from the total.

Traps

Accepting a fraction as the answer when asked which position a value sits at. A position must be a whole number. A fraction means that value is not in the pattern.
Putting $96$ in for $n$ in $T_{n} = 5 n + 1$ , when $96$ is the value. Only a position goes in for $n$ . If you know the value, set the rule equal to it.
Treating a remainder of $0$ as the first item of the cycle. A remainder of $0$ lands on the last item of the cycle, not the first.
Answering $30$ when the question asked how many desks are needed. Read what is wanted. Here $30$ is the seats and $14$ is the number of desks.

Learn and practise “Terms and positions” in the app →

Grade 10 Number patterns

What gets asked

You must be able to

Traps that cost marks

Worked example

Constant difference patterns

Rules to remember

Examples

Traps

Terms and positions

Rules to remember

Examples

Traps

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