This is the third of our Difficult Eleven Plus maths quizzes on Number Sequences. In the first two we asked you to find the missing term in a series. This quiz is more of a mixed bag. You’ll also be asked to find the rule for the n^{th} term which governs the sequence, and to create a sequence when you are given the rule.

In some number sequences the rules are very obvious. They might be increasing or decreasing by a certain amount each time. Others can be trickier, such as the n^{th} term squared or cubed. There are two ways to master number sequences: trial and error (try all the potential answers out for size – only one of them will work) and practise!

If you haven’t already played our previous quizzes on number sequences, then please go back and give them a try. The things you’ll learn by playing them will help you out, not just in this quiz, but in your exams when they come along. Practise makes perfect!

1.

Which sequence can be formed from the given rule for the n^{th} term?

n^{th} term = -n + 20

n

21, 22, 23, 24, ...

19, 18, 17, 16, ...

-21, -22, -23, -24, ...

-19, -18, -17, -16, ...

2.

What is the rule for the n^{th} term in the following sequence?

27, 26, 25, 24, …

27, 26, 25, 24, …

n^{th} term = -n + 28

n^{th} term = 9n

n^{th} term = -n + 30

n^{th} term = 7n + 20

To find the rule, test each option against the numbers in the sequence. Only one will work:

n = 1, and 27 = -1 + 28

n = 2, and 26 = -2 + 28

n = 3, and 25 = -3 + 28

n = 4, and 24 = -4 + 28

n = 1, and 27 = -1 + 28

n = 2, and 26 = -2 + 28

n = 3, and 25 = -3 + 28

n = 4, and 24 = -4 + 28

3.

Find the missing term.

-22, -26.5, X, -35.5, …

-22, -26.5, X, -35.5, …

-29.5

-30

-30.5

-31

The values are decreasing by 4.5 as the sequence continues:

-22 - 4.5 = -26.5

-26.5 - 4.5 = -31

-31 - 4.5 = -35.5, etc…

-22 - 4.5 = -26.5

-26.5 - 4.5 = -31

-31 - 4.5 = -35.5, etc…

4.

What is the rule for the n^{th} term in the following sequence?

8, 12, 16, 20, …

8, 12, 16, 20, …

n^{th} term = 5n + 2

n^{th} term = 8n

n^{th} term = 5n - 2

n^{th} term = 4n + 4

To find the rule, test each option against the numbers in the sequence. Only one will work:

n = 1, and 8 = (4 x 1) + 4

n = 2, and 12 = (4 x 2) + 4

n = 3, and 16 = (4 x 3) + 4

n = 4, and 20 = (4 x 4) + 4

n = 1, and 8 = (4 x 1) + 4

n = 2, and 12 = (4 x 2) + 4

n = 3, and 16 = (4 x 3) + 4

n = 4, and 20 = (4 x 4) + 4

5.

Find the missing term.

88, X , 122, 139, …

88, X , 122, 139, …

110

108

105

100

The values are increasing by 17 as the sequence continues:

88 + 17 = 105

105 + 17 = 122

122 = 17 = 139, etc…

88 + 17 = 105

105 + 17 = 122

122 = 17 = 139, etc…

6.

Find the missing term.

5.5, 6, X, 7, …

5.5, 6, X, 7, …

6.25

6.5

6.75

6.9

The sequence is increasing by 0.5 as it progresses:

5.5 = 0.5 = 6

6 + 0.5 = 6.5

6.5 + 0.5 = 7, etc…

5.5 = 0.5 = 6

6 + 0.5 = 6.5

6.5 + 0.5 = 7, etc…

7.

Which sequence can be formed from the given rule for the n^{th} term?

n^{th} term = 2.5n + 22

n

19.5, 22, 24.5, 27, ...

22.5, 25, 27.5, 30, ...

24.5, 27, 29.5, 32, …

25, 27.5, 30, 32.5, ...

The terms of the sequence are found by first putting n = 1, then n = 2, then n = 3 and finally n = 4 in the rule for the nth term = 2.5n + 22. As follows:

n = 1 gives (2.5 x 1) + 22 = 24.5

n = 2 gives (2.5 x 2) + 22 = 27

n = 3 gives (2.5 x 3) + 22 = 29.5

n = 4 gives (2.5 x 4) + 22 = 32

n = 1 gives (2.5 x 1) + 22 = 24.5

n = 2 gives (2.5 x 2) + 22 = 27

n = 3 gives (2.5 x 3) + 22 = 29.5

n = 4 gives (2.5 x 4) + 22 = 32

8.

Which sequence can be formed from the given rule for the n^{th} term?

n^{th} term = 4n - 17

n

-13, -9, -5, -1, …

-21, -25, -29, -33, ...

13, 9, 5, 1, ...

21, 25, 29, 33, ...

The terms of the sequence are found by first putting n = 1, then n = 2, then n = 3 and finally n = 4 in the rule for the nth term = 4n - 17. As follows:

n = 1 gives (4 x 1) - 17 = -13

n = 2 gives (4 x 2) - 17 = -9

n = 3 gives (4 x 3) - 17 = -5

n = 4 gives (4 x 4) - 17 = -1

n = 1 gives (4 x 1) - 17 = -13

n = 2 gives (4 x 2) - 17 = -9

n = 3 gives (4 x 3) - 17 = -5

n = 4 gives (4 x 4) - 17 = -1

9.

What is the rule for the n^{th} term in the following sequence?

-1, -2, -3, -4, …

-1, -2, -3, -4, …

n^{th} term = n - n

n^{th} term = n - 2n

n^{th} term = -n^{2}

n^{th} term = -n - n

To find the rule, test each option against the numbers in the sequence. Only one will work:

n = 1, and -1 = 1 - (2 x 1)

n = 2, and -2 = 2 - (2 x 2)

n = 3, and -3 = 3 - (2 x 3)

n = 4, and -4 = 4 - (2 x 4)

Of course, -n would work too, but that wasn’t one of the options!

n = 1, and -1 = 1 - (2 x 1)

n = 2, and -2 = 2 - (2 x 2)

n = 3, and -3 = 3 - (2 x 3)

n = 4, and -4 = 4 - (2 x 4)

Of course, -n would work too, but that wasn’t one of the options!

10.

Find the missing term.

0.5, 2, 4.5, X, 12.5, 18, …

0.5, 2, 4.5, X, 12.5, 18, …

6.5

7.5

8

10

This one was very tricky, so well done if you got it right.

The rule for this sequence is the n^{th} term = n^{2} ÷ 2:

1^{2} ÷ 2 = 0.5

2^{2} ÷ 2 = 2

3^{2} ÷ 2 = 4.5

4^{2} ÷ 2 = 8

5^{2} ÷ 2 = 12.5

6^{2} ÷ 2 = 18, etc…

The rule for this sequence is the n

1

2

3

4

5

6

n = 1 gives -1 + 20 = 19

n = 2 gives -2 + 20 = 18

n = 3 gives -3 + 20 = 17

n = 4 gives -4 + 20 = 16