 # Find the nth Term of Quadratic Sequences

In this worksheet, students practise finding the nth term of a quadratic sequence. 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

The nth term of a quadratic has the form an2 + bn + c and to find it, we need to find the values for a, b and c

It is quite useful to notice that an nth term is made up of two distinct parts.

an2        +          bn + c

Finding the nth term of a quadratic is quite easy, if not straightforward. Just follow these steps and you won't go far wrong.

Example: Find the nth term for the sequence 5, 15, 29, 47

Step 1: Finding a:

The value of a will always be half the second difference

The difference between the first and second terms is 10

The difference between the second and third terms is 14

The difference between the third and fourth  terms is 18

We can see that the second difference is 4, so the value of a is 2

Step 2: Set up a table

 Position 1 2 3 4 an2 = 2n2 bn+c Target 5 15 29 47

In this, the position is just the value of n (where it is in the sequence)

We have just found a so we can fill in the second line by substituting values of  n into 2n2.

 Position 1 2 3 4 an2 = 2n2 2 8 18 32 bn+c Target 5 15 29 47

Step 3: Find the value of bn + c

In our table, we now know what the quadratic elelment is giving us, so we can now find out what the linear components will be. These are just the differences between an2 and the target

 Position 1 2 3 4 an2 = 2n2 2 8 18 32 bn+c 3 7 11 15 Target 5 15 29 47

We can now see that these numbers in red (3, 7, 11, 15) form a linear sequence with an nth term of 4n - 1

Step 4: Put it all together

Earlier, we said that...

an2        +          bn + c

2n2     +    4n - 1

(Quadratic Sequence: an2 + bn +c)

What is the value of a for the quadratic sequence 6, 15, 28, 45...

(Quadratic Sequence: an2 + bn +c)

What are the values of b and c for the quadratic seuqence...

6, 15, 28, 45...

complete the following sentence...

What are the first 4 terms for the sequence 2n2 - n + 3?

Match the sequence with the nth term.

## Column B

4, 9, 16, 25
4(n2 - 2n)
-1, 6, 17, 32
n2 + 2n + 1
2, 9, 22, 41, 66
2n2 2n - 4
-4, 0, 12, 32, 60
3n2 - 2n + 1

Which of these is the correct th term for...

0, 4, 14, 30...

3n2 - 5n + 2

5n2 - 3n + 2

2n2 - 5n + 3

2n2 - 3n + 5

What are the first 5 terms for the sequence 3n2 + 2n + 1?

3n2 - 5n + 2

5n2 - 3n + 2

2n2 - 5n + 3

2n2 - 3n + 5

Find the nth term for 3, 8, 15, 24, 35 in the form an2 + bn + c.

3n2 - 5n + 2

5n2 - 3n + 2

2n2 - 5n + 3

2n2 - 3n + 5

Find the nth term for 4, 13, 26, 43, 64 in the form an2 + bn + c.

3n2 - 5n + 2

5n2 - 3n + 2

2n2 - 5n + 3

2n2 - 3n + 5

Find the nth term for 6, 3, -4, -15, -36 in the form an2 + bn + c.

3n2 - 5n + 2

5n2 - 3n + 2

2n2 - 5n + 3

2n2 - 3n + 5

• Question 1

(Quadratic Sequence: an2 + bn +c)

What is the value of a for the quadratic sequence 6, 15, 28, 45...

2
EDDIE SAYS
We find a common second difference here of 4, this makes the value of a....
• Question 2

(Quadratic Sequence: an2 + bn +c)

What are the values of b and c for the quadratic seuqence...

6, 15, 28, 45...

EDDIE SAYS
We worked out before that a was 2. If we take away 2n2 from the sequence we are looking for, we get the sequence 4, 7, 10, 13... This gives us the sequence 3n + 1
• Question 3

complete the following sentence...

EDDIE SAYS
Remember; a - Half the second difference b and c - use a comparison table
• Question 4

What are the first 4 terms for the sequence 2n2 - n + 3?

EDDIE SAYS
To find the numbers in a sequence, we just put n into the nth term. So for the first term, n=1 and we plug that into the nth term. 2 x (1),sup>2 - (1) + 3 = 4 Repeat ad infinitum (or as long as you need to)
• Question 5

Match the sequence with the nth term.

## Column B

4, 9, 16, 25
n2 + 2n + 1
-1, 6, 17, 32
2n2 2n - 4
2, 9, 22, 41, 66
3n2 - 2n + 1
-4, 0, 12, 32, 60
4(n2 - 2n)
EDDIE SAYS
You can match a couple of these quickly. There is only one option for 4(n2 - 2n) as there is only one that has all the numbers in the 4 times tables. To match 2n2 2n - 4, we have a negative at the end, which one of the ones that are left have a negative For the other two, consider which one will have the larger numbers?
• Question 6

Which of these is the correct th term for...

0, 4, 14, 30...

3n2 - 5n + 2
EDDIE SAYS
I love questions like this, you don't have to work out the whole thing. If you start by looking at the differences and second differences, you find the second difference is 6. With the value off a being half of this, a = 3 and there is only one option that fits.
• Question 7

What are the first 5 terms for the sequence 3n2 + 2n + 1?

EDDIE SAYS
To find the numbers in a sequence, we just put n into the nth term. So for the first term, n=1 and we plug that into the nth term. 3 x (1),sup>2 + 2(1) + 1 = 6 Repeat ad infinitum (or as long as you need to)
• Question 8

Find the nth term for 3, 8, 15, 24, 35 in the form an2 + bn + c.

EDDIE SAYS
The second difference comes out as 2 which makes a =1 Using the table of values, we should get the linear expression 2, 4, 6, 8 This means that b = 2 and c = 0
• Question 9

Find the nth term for 4, 13, 26, 43, 64 in the form an2 + bn + c.

EDDIE SAYS
The second difference comes out as 4 which makes a =2 Using the table of values, we should get the linear expression 2, 5, 8, 11 This means that b = 3 and c = -1
• Question 10

Find the nth term for 6, 3, -4, -15, -36 in the form an2 + bn + c.

EDDIE SAYS
The second difference comes out as -4 which makes a =-2 Using the table of values, we should get the linear expression 8, 11, 14, 17 This means that b = 3 and c = 5
---- OR ----

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