Welcome to the third of our Medium level Eleven Plus maths quizzes on Number Sequences. In the first two we introduced you to the n^{th} term and asked you to create a sequence of numbers from it. In this one we turn the tables. We shall give you a sequence of numbers and ask you to find the rule for the n^{th} term which created it.

These can be a bit tricky. It might be obvious that the numbers are going up by 4 each time, but the rule won’t be “n + 4”. That rule would give you the following sequence:

5, 6, 7, 8, … (1 + 4 = 5, 2 + 4 = 6, 3 + 4 = 7, 4 + 4 = 8, etc.)

The best advice I can give you is to look at the four options you are presented with, and to try them out. Only one of the answers will work for all the numbers in the sequence. It’s your job to find out which!

1.

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

2, 6, 10, 14, …

2, 6, 10, 14, …

n^{th} term = -n + 3n

n^{th} term = n + 4

n^{th} term = 4n - 2

n^{th} term = 2n

2.

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

2.5, 5, 7.5, 10, …

2.5, 5, 7.5, 10, …

n^{th} term = n + 2.5

n^{th} term = 2n + n

n^{th} term = 5n - 2

n^{th} term = 2.5n

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

n = 1, and 2.5 = 1 x 2.5

n = 2, and 5 = 2 x 2.5

n = 3, and 7.5 = 3 x 2.5

n = 4, and 10 = 4 x 2.5

n = 1, and 2.5 = 1 x 2.5

n = 2, and 5 = 2 x 2.5

n = 3, and 7.5 = 3 x 2.5

n = 4, and 10 = 4 x 2.5

3.

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

10, 13, 16, 19, …

10, 13, 16, 19, …

n^{th} term = 3n + 7

n^{th} term = n + 10

n^{th} term = 2n + 8

n^{th} term = 10n - 7

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

n = 1, and 10 = (3 x 1) + 7

n = 2, and 13 = (3 x 2) + 7

n = 3, and 16 = (3 x 3) + 7

n = 4, and 19 = (3 x 4) + 7

n = 1, and 10 = (3 x 1) + 7

n = 2, and 13 = (3 x 2) + 7

n = 3, and 16 = (3 x 3) + 7

n = 4, and 19 = (3 x 4) + 7

4.

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

1, 4, 9, 16, …

1, 4, 9, 16, …

n^{th} term = n^{3}

n^{th} term = 3n - 3

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

n^{th} term = 4n - 2

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

n = 1, and 1 = 1 x 1, or 1^{2}

n = 2, and 4 = 2 x 2, or 2^{2}

n = 3, and 9 = 3 x 3, or 3^{2}

n = 4, and 16 = 4 x 4, or 4^{2}

n = 1, and 1 = 1 x 1, or 1

n = 2, and 4 = 2 x 2, or 2

n = 3, and 9 = 3 x 3, or 3

n = 4, and 16 = 4 x 4, or 4

5.

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

9, 15, 21, 27, …

9, 15, 21, 27, …

n^{th} term = n + 8

n^{th} term = 6n + 3

n^{th} term = 9n

n^{th} term = 7n + 2

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

n = 1, and 9 = (6 x 1) + 3

n = 2, and 15 = (6 x 2) + 3

n = 3, and 21 = (6 x 3) + 3

n = 4, and 27 = (6 x 4) + 3

n = 1, and 9 = (6 x 1) + 3

n = 2, and 15 = (6 x 2) + 3

n = 3, and 21 = (6 x 3) + 3

n = 4, and 27 = (6 x 4) + 3

6.

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

-6, -7, -8, -9, …

-6, -7, -8, -9, …

n^{th} term = -n x 6

n^{th} term = -n - 6

n^{th} term = -n - 5

n^{th} term = n - 6

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

n = 1, and -6 = -1 - 5

n = 2, and -7 = -2 - 5

n = 3, and -8 = -3 - 5

n = 4, and -9 = -4 - 5

n = 1, and -6 = -1 - 5

n = 2, and -7 = -2 - 5

n = 3, and -8 = -3 - 5

n = 4, and -9 = -4 - 5

7.

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

5, 13, 21, 29, …

5, 13, 21, 29, …

n^{th} term = n^{2} + 5

n^{th} term = 5n + 3

n^{th} term = 7n - 2

n^{th} term = 8n - 3

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

n = 1, and 5 = (8 x 1) - 3

n = 2, and 13 = (8 x 2) - 3

n = 3, and 21 = (8 x 3) - 3

n = 4, and 29 = (8 x 4) - 3

n = 1, and 5 = (8 x 1) - 3

n = 2, and 13 = (8 x 2) - 3

n = 3, and 21 = (8 x 3) - 3

n = 4, and 29 = (8 x 4) - 3

8.

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

5, 4, 3, 2, …

5, 4, 3, 2, …

n^{th} term = 6n - 1

n^{th} term = -n + 6

n^{th} term = 5n

n^{th} term = -n + 7

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

n = 1, and 5 = -1 + 6

n = 2, and 4 = -2 + 6

n = 3, and 3 = -3 + 6

n = 4, and 2 = -4 + 6

n = 1, and 5 = -1 + 6

n = 2, and 4 = -2 + 6

n = 3, and 3 = -3 + 6

n = 4, and 2 = -4 + 6

9.

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

7, 14, 21, 28, …

7, 14, 21, 28, …

n^{th} term = 7n

n^{th} term = 6n + 1

n^{th} term = 8n - 1

n^{th} term = n^{7}

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

n = 1, and 7 = 1 x 7

n = 2, and 14 = 2 x 7

n = 3, and 21 = 3 x 7

n = 4, and 28 = 4 x 7

n = 1, and 7 = 1 x 7

n = 2, and 14 = 2 x 7

n = 3, and 21 = 3 x 7

n = 4, and 28 = 4 x 7

10.

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

4, 6, 8, 10, …

4, 6, 8, 10, …

n^{th} term = 4n

n^{th} term = 2n + 2

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

n^{th} term = 2n - 2

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

n = 1, and 4 = (2 x 1) + 2

n = 2, and 6 = (2 x 2) + 2

n = 3, and 8 = (2 x 3) + 2

n = 4, and 10 = (2 x 4) + 2

n = 1, and 4 = (2 x 1) + 2

n = 2, and 6 = (2 x 2) + 2

n = 3, and 8 = (2 x 3) + 2

n = 4, and 10 = (2 x 4) + 2

n = 1, and 2 = (4 x 1) - 2

n = 2, and 6 = (4 x 2) - 2

n = 3, and 10 = (4 x 3) - 2

n = 4, and 14 = (4 x 4) - 2