 # Find the Nth Term of Complex Quadratic Sequences

In this worksheet, students find expressions which represent quadratic sequences (in the form an² + bn + c) and apply these 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

The nth term of a quadratic sequence is expressed in the form an2 + bn + c.

To find the correct expression, 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

Although this may look daunting, finding the nth term of a quadratic sequence can be quite straightforward.

Just follow the steps laid about below and you won't go astray.

e.g. Find the nth term for this sequence: 5, 15, 29, 47, ...

Step 1: Find a

There is a golden rule we must apply here: 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 (4 × 0.5).

Step 2: Set up a table to support you

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

In this table, the 'Position' is just the value of n (i.e. where it is in the sequence).

We have just found the value of 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 the value of the quadratic element, so we can now us this to find the linear element.

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 blue (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

There are many steps here but, if we follow them systematically, we will reach an accurate quadratic expression each time.

In this activity, we will find quadratic expressions which represent sequences and apply these to find unknown terms.

As these methods can be long and complex, it is a good idea to have a pen and paper handy so you can compare your working to what our maths teacher has written as an example.

Quadratic Sequence: an2 + bn + c

What is the value of a in quadratic expression of this sequence?

6, 15, 28, 45, ...

Quadratic Sequence: an2 + bn + c

Following on from the previous question, what is the value of b and c in quadratic expression of this sequence?

6, 15, 28, 45, ...

Complete the sentences below to summarise the process we have just applied.

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

Match each sequence below with the correct expression of its nth term.

## Column B

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

Which of the options below is the correct nth term expression for this sequence?

0, 4, 14, 30, ...

3n2 - 5n + 2

5n2 - 3n + 2

2n2 - 5n + 3

2n2 - 3n + 5

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

3n2 - 5n + 2

5n2 - 3n + 2

2n2 - 5n + 3

2n2 - 3n + 5

Find the nth term for the sequence 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 the sequence 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 the sequence 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 in quadratic expression of this sequence?

6, 15, 28, 45, ...

2
EDDIE SAYS
Remember our golden rule: "The value of a will always be half the second difference." 1st difference: 9 (15 - 6), 13 (28 - 15), 17 (45 - 28), etc. 2nd difference: 13 - 9 / 17 - 3 = 4 0.5 × 4 = 2 Did you apply that golden rule successfully? Let's move on to find the values of b and c in this expression now...
• Question 2

Quadratic Sequence: an2 + bn + c

Following on from the previous question, what is the value of b and c in quadratic expression of this sequence?

6, 15, 28, 45, ...

EDDIE SAYS
We worked out in the previous question that: a = 2 If we take 2n2 from the sequence we are looking for, we get the revised sequence: 4, 7, 10, 13, ... Can you spot the relationship between these numbers? The position n is being multiplied by 3 then 1 is added, which we express as: 3n + 1 So b = 3 and c = 1.
• Question 3

Complete the sentences below to summarise the process we have just applied.

EDDIE SAYS
Remember, we have to halve the second difference to find a. Then we can draw a table to help us investigate the relationship between the quadratic (n2) and linear elements (bn + c). How did you get on applying this process in the last two questions?
• Question 4

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

EDDIE SAYS
To find the numbers in a sequence, we just need to substitute n in our expression for each term, one at a time. So for the first term, n = 1: 2 × 12 - 1 + 3 = 4 For the second term, n = 2: 2 × 22 - 2 + 3 = 9 For the third term, n = 3: 2 × 32 - 3 + 3 = 18 For the fourth term, n = 4: 2 × 42 - 4 + 3 = 31 You can continue to repeat this process for as many terms as you require, or for a specific term if that is asked for too.
• Question 5

Match each sequence below with the correct expression of its nth term.

## Column B

4, 9, 16, 25, ...
n2 + 2n + 1
-1, 6, 17, 32, ...
2n2 + 2n - 4
2, 9, 22, 41, ...
3n2 - 2n + 1
-4, 0, 12, 32, ...
4(n2 - 2n)
EDDIE SAYS
We can match a couple of these options quickly. There is only one option for 4(n2 - 2n), as there is only one sequence which contains numbers from the 4 times tables. With 2n2 + 2n - 4, we have a negative at the end. Which other sequence involves negative numbers? For the remaining two expressions, let's consider which one will have the larger numbers. 3n2 - 2n + 1 will generate larger numbers, so we should match that with the sequence containing the highest numbers. Were you able to find all the pairs using those top tips? Alternatively, you could have calculated each one separately using our rules, but this would probably have taken you a lot more time so it is always a good idea to look for shortcuts if possible.
• Question 6

Which of the options below is the correct nth term expression for this sequence?

0, 4, 14, 30, ...

3n2 - 5n + 2
EDDIE SAYS
Did you spot that we didn't have to work out the whole thing here? If we start by looking at the differences and second differences, we find that the second difference is 6: 1st differences: + 4, + 10, + 16, etc. 2nd differences: + 6, + 6, etc. With the value of a being half of this, we know that: a = 3 There is therefore only one option that fits, as our correct answer needs to have a 3n2 in it. Did you spot that?
• Question 7

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

EDDIE SAYS
To find the terms in a sequence, we just need to plug in values for n and apply the expression. So for the first term, n = 1: 3 × 12 + (2 × 1) + 1 = 6 For the second term, n = 2: 3 × 22 + (2 × 2) + 1 = 17 For the third term, n = 3: 3 × 32 + (2 × 3) + 1 = 34 Can you follow this process to find the final two terms independently?
• Question 8

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

EDDIE SAYS
Let's find a to start. The second difference here is 2, which means a = 1. Using the table of values, we should reach the revised linear expression: 2, 4, 6, 8, etc. If n = 1, then (b × 1) + c = 2 If n = 2, then (b × 2) + c = 4 If n = 3, then (b × 3) + c = 6, etc. This means that b = 2 and c = 0. If we put this all together, we reach: 1n2 + 2n + 0 Which could be simplified to: n2 + 2n
• Question 9

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

EDDIE SAYS
The second difference here is 4 which means a = 2. Using the table of values, we should reach the revised linear expression: 2, 5, 8, 11, etc. If n = 1, then (b × 1) + c = 2 If n = 2, then (b × 2) + c = 5 If n = 3, then (b × 3) + c = 8, etc. This means that b = 3 and c = -1. If we put this all together, we reach: 2n2 + 3n - 1 Just one more challenge remaining now!
• Question 10

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

EDDIE SAYS
The second difference here is -4 which means a = -2. Using the table of values, we should reach the revised linear expression: 8, 11, 14, 17, etc. If n = 1, then (b × 1) + c = 8 If n = 2, then (b × 2) + c = 11 If n = 3, then (b × 3) + c = 14, etc. This means that b = 3 and c = 5. If we put this all together, we reach: -2n2 + 3n + 5 Great work completing this activity - this was not an easy one!
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