Number sequences illustration | A chain
Numbers sequences are just chains of numbers put together by certain rules.

Number Sequences 3 (Difficult)

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 nth 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 nth 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.
What is the rule for the nth term in the following sequence?
27, 26, 25, 24, …
nth term = -n + 28
nth term = 9n
nth term = -n + 30
nth 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
2.
Which sequence can be formed from the given rule for the nth term?
nth term = 4n - 17
-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
3.
Find the missing term.
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…
4.
Find the missing term.
-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…
5.
Which sequence can be formed from the given rule for the nth term?
nth term = 2.5n + 22
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
6.
Find the missing term.
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…
7.
Find the missing term.
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 nth term = n2 ÷ 2:
12 ÷ 2 = 0.5
22 ÷ 2 = 2
32 ÷ 2 = 4.5
42 ÷ 2 = 8
52 ÷ 2 = 12.5
62 ÷ 2 = 18, etc…
8.
What is the rule for the nth term in the following sequence?
-1, -2, -3, -4, …
nth term = n - n
nth term = n - 2n
nth term = -n2
nth 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!
9.
What is the rule for the nth term in the following sequence?
8, 12, 16, 20, …
nth term = 5n + 2
nth term = 8n
nth term = 5n - 2
nth 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
10.
Which sequence can be formed from the given rule for the nth term?
nth term = -n + 20
21, 22, 23, 24, ...
19, 18, 17, 16, ...
-21, -22, -23, -24, ...
-19, -18, -17, -16, ...
the rule for the nth term = -n + 20. As follows:
n = 1 gives -1 + 20 = 19
n = 2 gives -2 + 20 = 18
n = 3 gives -3 + 20 = 17
n = 4 gives -4 + 20 = 16
Author:  Frank Evans

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