 # Find the Nth Term for Quadratic Sequences

In this worksheet, students will apply knowledge of square numbers to find the nth term for quadratic sequences and apply these rules to find unknown terms. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Algebra

Curriculum subtopic:   Sequences

Difficulty level:   ### QUESTION 1 of 10

Finding the nth term means that we don't need to continuously apply a rule to the previous term in a sequence to find a missing term.

When we are expressing such a sequence in the terms of an unknown variable (n), the nth term will contain the term n2.

Whilst there is a 'correct' way of identifying such a sequence, sometimes it is easier to see if the sequence is related to the sequence of square numbers which we are familiar with (i.e. 1, 4, 9, 16, etc.).

If this 'quicker' approach is expected in an exam, the question will usually carry less marks (like 1 or 2).

Let's look at some examples now to see how we can use our knowledge of square numbers to help us with these sequences.

e.g. Find the nth term of this sequence: 2, 18, 18, 32, ...

Let's work from a sequence which we know uses the n2 term, our square numbers: 1, 4, 9, 16, etc.

Can you see any relationship between these numbers and our starting sequence?

Our new sequence is doubling these square numbers.

This makes our nth term for this sequence: 2n2

e.g. Find the nth term of this sequence: 3, 7, 11, 18, ...

Again, we can compare this sequence with our square numbers.

You may notice that our sequence is just two more than this series.

So our sequence in this case would be: n2 + 2

e.g. Find the nth term of this sequence: 19, 16, 11, 4, ...

If we compare this to our square numbers, you may notice that they add up to 20.

This could be looked at like this: "If we take the square numbers away from 20, we reach our target."

This makes our nth term in this case: 20 - n2

In this activity, we will apply our knowledge of square numbers to find the nth term for quadratic sequences and apply these rules to find unknown terms.

If you are not feeling totally confident with your square numbers, you may want to revise these before you attempt this activity.

What are the next two numbers in this sequence?

1, 4, 9, 16, ...

What is the 6th number in this sequence?

3, 12, 27, 48, ...

What is the next number in this sequence?

-1, 2, 7, 14, ...

25

27

23

Match each sequence below with its correct nth term.

## Column B

1, 4, 9, 16, ...
n2 + 5
3, 12, 27, 48, ...
3n2
6, 9, 14, 21, ...
n2

Consider this sequence:

3, 12, 27, 48, ...

The nth term for this sequence can be written as an2.

Type the value of a in the box below.

Which of the numbers listed below will be in the sequence 0, 3, 8, 15, ...?

24

80

50

37

63

120

Consider this sequence:

-1, -4, -9, -16, ...

The nth term for this sequence can be written as an2.

Type the value of a in the box below.

9, 6, 1, -6, ...

The nth term for this sequence can be written as a - bn2.

Type the values of a and b in the spaces below.

What is the only number that appears in the first ten terms in both the sequences defined by n2 and 2n - 1?

What is the nth term for this sequence?

4, 7, 12, 19, ...

n2 + 3

3n2

3 - n2

• Question 1

What are the next two numbers in this sequence?

1, 4, 9, 16, ...

EDDIE SAYS
Did you spot that this was our much-used sequence of square numbers? A square number is the product of a number multiplied by itself, e.g. 3 × 3 = 9, 4 × 4 = 16, etc.
• Question 2

What is the 6th number in this sequence?

3, 12, 27, 48, ...

108
EDDIE SAYS
If we compare the numbers in this sequence to the square numbers, we notice that they are 3 times bigger. So our nth term here will be generated using 3n2. The 6th number will, therefore, be: 3 × 62 = 3 × 36 = 108
• Question 3

What is the next number in this sequence?

-1, 2, 7, 14, ...

23
EDDIE SAYS
If we compare this sequence to the square numbers, we can see they are 2 smaller. So our rule here is n2 - 2. Therefore, the answer we are looking for will be 2 less than the 5th square number: 25 - 2 = 23
• Question 4

Match each sequence below with its correct nth term.

## Column B

1, 4, 9, 16, ...
n2
3, 12, 27, 48, ...
3n2
6, 9, 14, 21, ...
n2 + 5
EDDIE SAYS
We need to compare each of these sequences to the square numbers one at a time. n2 means we are looking for a sequence of numbers which are the same as the square numbers. Can you spot those? n2 + 5 means we are looking for a sequence of numbers which are 5 more than the square numbers. 3n2 means we are looking for a sequence of numbers which are 3 times the square numbers.
• Question 5

Consider this sequence:

3, 12, 27, 48, ...

The nth term for this sequence can be written as an2.

Type the value of a in the box below.

3
EDDIE SAYS
Can you see a relationship between the numbers in this sequence and the square numbers? All of the numbers we are looking at are 3 times larger than the square numbers. To express this as a rule, we would write: 3n2
• Question 6

Which of the numbers listed below will be in the sequence 0, 3, 8, 15, ...?

24
80
63
120
EDDIE SAYS
The numbers in our sequence here are 1 less than the square numbers. We can use this to work out that any number which is one less than a square number will be in this sequence. Can you identify these numbers in the list?
• Question 7

Consider this sequence:

-1, -4, -9, -16, ...

The nth term for this sequence can be written as an2.

Type the value of a in the box below.

-
-1
- 1
EDDIE SAYS
Did you notice that these are just the square numbers with a minus sign in front of them? Therefore, we would express the nth term for this sequence as: - n2 When we just have the variable itself in an expression, we know this means the same as 1 × n. So the value of a is 1.
• Question 8

9, 6, 1, -6, ...

The nth term for this sequence can be written as a - bn2.

Type the values of a and b in the spaces below.

EDDIE SAYS
This is more challenging, as we need to notice that these numbers combine with the square numbers to give an overall value of 10. So to find the nth term, we have to take the square numbers away from 10, which is expressed like this: 10 - n2 Again, there is no numerical value attached to n, so we need to remember that this is the same as 1 × n. Did you type those values into the spaces accurately?
• Question 9

What is the only number that appears in the first ten terms in both the sequences defined by n2 and 2n - 1?

1
EDDIE SAYS
The easiest way to find this answer is to write out the two sequences up to the 10th term: n2: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc. 2n - 1: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, etc. The only number which appears in the first 10 terms in both sequences is, therefore, 1.
• Question 10

What is the nth term for this sequence?

4, 7, 12, 19, ...

n2 + 3
EDDIE SAYS
Let's compare this final sequence to the square numbers. Can you see that each of these terms is 3 more than the square numbers? We express this as: n2 + 3 Fantastic work completing this activity! Consider moving on to try the Level 3 activity linked to this one which focuses on more complex quadratic sequences - good luck if you choose to give this a try!
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