# Find and Apply Rules in Special Numbers

In this worksheet, students will identify missing numbers in sequences of prime, square and triangle numbers.

Key stage:  KS 2

Curriculum topic:   Verbal Reasoning

Curriculum subtopic:   Number Sequences

Difficulty level:

### QUESTION 1 of 10

What great timing number detective, we’ve found a new puzzle and we need your help!

Can you work out which number comes next in this sequence?

2, 3, 5, 7, __

The missing number is 11.

This is a sequence of prime numbers and 11 is the next prime number after 7.

Prime numbers are numbers that can only be divided by 1 and themselves.

What type of special numbers make the following sequence and which number would come next?

1, 4, 9, 16, __

The numbers in this sequence are square numbers and the next square number is 25, which is 5 x 5 or 5².

Square numbers are the answers once you have multiplied a number by itself, such as:

1²= 1 x 1= 1

2² = 2 x 2 = 4

3² = 3 x 3= 9

4² = 4 x 4= 16

5² = 5 x 5= 25

Here is a sequence of numbers called triangle numbers. Can you work out the pattern?

1, 3, 6, 10, 15, 21

In this sequence, the amount added each time is going up by one.

The next number in this sequence would be 28 as we need to add 7 next to 21.

It is now your mission to identify the missing numbers in sequences like the ones that we have just looked at.

Be sure to look out for prime numbers, square numbers and triangle numbers.

Write down the relationship between the numbers as you go, it will make it easier to spot a pattern.

Good luck number detective!

We are counting on you...

What type of number sequence is the sequence below?

2, 3, 5, 7, 11

Square number sequence

Prime number sequence

Triangle number sequence

What type of number sequence is this?

1, 4, 9, 16, 25

Prime number sequence

Square number sequence

Triangle number sequence

Which number would come next in the following sequence?

3, 5, 7, 11, __

Which number would come next in the following sequence?

7, 11, 13, 17, __

Which number would come next in the following sequence?

Fill in the blank below.

Which number would come next in the following sequence?

Which number would come next in the following sequence?

1, 3, 6, 10, 15, __

Which number would come next in the following sequence?

Match the sequences below to their sequence types.

## Column B

7, 11, 13, 17, 19
Triangle numbers
64, 81, 100, 121, 144
Prime numbers
1, 3, 6, 10, 15
Square numbers

Match the sequences below to the next number in each.

## Column B

11, 13, 17, 19,
25
1, 3, 6, 10,
15
1, 4, 9, 16,
23
• Question 1

What type of number sequence is the sequence below?

2, 3, 5, 7, 11

Prime number sequence
EDDIE SAYS
Hey number detective! Did you notice that all of the numbers in the sequence were prime numbers? Prime numbers only have 2 factors: one and themselves. This means that they aren't in any other times tables apart from the 1 times table and their own times table. What is the next prime number after 11?
• Question 2

What type of number sequence is this?

1, 4, 9, 16, 25

Square number sequence
EDDIE SAYS
All of the numbers in this sequence are square numbers. Square numbers are made when a number multiples by itself. The next square number in this sequence is 36, which is 6 x 6.
• Question 3

Which number would come next in the following sequence?

3, 5, 7, 11, __

13
EDDIE SAYS
Did you notice that this was a sequence of prime numbers? Prime numbers can ONLY be divided by 1 and themselves. The next prime number after 11 is 13. We know it is a prime because the number 13 is only in the 1 and 13 times tables. It is worth spending time to learn your prime numbers off by heart to help you become a superstar number detective!
• Question 4

Which number would come next in the following sequence?

7, 11, 13, 17, __

23
EDDIE SAYS
It's prime time again! All of the numbers in this sequence are prime numbers. The next prime number after 19 is 23. 23 only has 2 factors: 1 and 23. This makes it a prime number.
• Question 5

Which number would come next in the following sequence?

Fill in the blank below.

EDDIE SAYS
Did you spot the square number sequence here? 16 is 4 squared so the next number in the sequence needs to be the answer to 5 squared. 5 x 5 = 25. 25 is the next square number after 16.
• Question 6

Which number would come next in the following sequence?

EDDIE SAYS
It's more square numbers! If we see any more we'll turn into squares! This sequence is going up in square numbers, 100 is 10 squared so we need to work out the answer to 11 squared. 11 x 11 = 121.
• Question 7

Which number would come next in the following sequence?

1, 3, 6, 10, 15, __

21
EDDIE SAYS
We have some triangle numbers here. The amount you add in a triangle number sequence increases by one each time: We need to add 6 to 15. This makes 21 the next number in the sequence.
• Question 8

Which number would come next in the following sequence?

EDDIE SAYS
Did you notice that the numbers were getting smaller each time here? This is a square number sequence that is DECREASING so we need to work out the square number that comes BEFORE 81. 81 is 9 squared so we need to work out 8 squared. 8 x 8 = 64 I ate and I ate until I was sick on the floor, eight times eight is sixty-four!
• Question 9

Match the sequences below to their sequence types.

## Column B

7, 11, 13, 17, 19
Prime numbers
64, 81, 100, 121, 144
Square numbers
1, 3, 6, 10, 15
Triangle numbers
EDDIE SAYS
How many sequences did you recognise? Prime numbers often stand out as we aren't used to seeing them in our times tables. Try and learn all of your square numbers up to 144 off by heart to help you achieve superstar detective status!
• Question 10

Match the sequences below to the next number in each.

## Column B

11, 13, 17, 19,
23
1, 3, 6, 10,
15
1, 4, 9, 16,
25
EDDIE SAYS
How many matches did you find? The 1st sequence uses prime numbers. The next prime number after 19 is 23. The 2nd sequence uses triangle numbers. The next triangle number after 10 is 15 as we need to + 5. The 3rd sequence uses square numbers. The next square number after 16 is 25. Excellent sequencing skills detective!
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