Quadratic sequences

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Lesson details

Learning outcome

I can find the n^th term formula for a quadratic sequence.

Key learning points

A quadratic sequence does not have a common difference.
The second differences between terms are equal.
The general n^th term formula is an^2 + bn + c

Keywords

N^th term - The nth term of a sequence is the position of a term in a sequence where n stands for the term number.
Quadratic - A quadratic is an equation, graph, or sequence whereby the highest exponent of the variable is 2

Common misconception

Since there is no common difference, it is impossible to find the n^th term.

The first learning cycle shows how we can build quadratic sequences from n^2 and a linear sequence.

Teacher tip

The first learning cycle can be extended and pupils can create lots of different quadratic sequences if you feel that building quadratic sequences needs more exploration before moving to finding the n^th term for any quadratic sequence.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0

except where otherwise stated. See Oak's terms & conditions

(Collection 2).

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Prior knowledge starter quiz

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6 Questions

Q1.
Which of these could be the first 5 terms of a quadratic sequence?

Correct answer: 5, 8, 12, 17, 23, ...

8, 10, 14, 22, 38, ...

Correct answer: 10, 14, 15, 13, 8, ...

11, 20, 31, 51, 82, ...

Q2.
Match the first 5 terms of these quadratic sequences to their common second differences.

Correct Answer:-5, -3, 0, 4, 9, ...,common second difference of +1

common second difference of +1

Correct Answer:-1, 7, 13, 17, 19, ...,common second difference of -2

common second difference of -2

Correct Answer:4, 5, 8, 13, 20, ...,common second difference of +2

common second difference of +2

Correct Answer:7, 11, 14, 16, 17, ...,common second difference of -1

common second difference of -1

Q3.
Which is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts: 3, 6, 11, 18, 27, ...?

$$n^2 -3$$

$$n^2 -2$$

$$n^2 -1$$

Correct answer: $$n^2 + 2$$

$$n^2 + 3$$

Q4.
The quadratic sequence which starts: 13, 16, 21, 28, ... has $$n^\text{th}$$ term rule $$n^2 +$$

Correct Answer: 12

Q5.
What is the $$n^\text{th}$$ term of the linear sequence which starts -1, -2, -3, -4, ... ?

$$1-2n$$

Correct answer: $$-n$$

$$n-1$$

$$n-2$$

$$n^2 -2$$

Q6.
What is the $$n^\text{th}$$ term of the linear sequence which starts -1, 2, 5, 8, ... ?

$$-n + 3$$

$$3-4n$$

$$1-3n$$

Correct answer: $$3n -4$$

$$4n -3$$

Assessment exit quiz

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6 Questions

Q1.
Which two sequences from the options below could be added to make the quadratic sequence which starts: 3, 7, 13, 21, ...?

Correct answer: $$n^2: 1, 4, 9, 16, ...$$

$$2n^2: 2, 8, 18, 32, ...$$

$$-2n+3 : 1, -1, -3, -5, ...$$

Correct answer: $$n+1 : 2, 3, 4, 5, ...$$

$$3n-2: 1, 4, 7, 10, ...$$

Q2.
Which two sequences from the options below could be added to make the quadratic sequence which starts: 3, 7, 15, 27, ...?

$$n^2: 1, 4, 9, 16, ...$$

Correct answer: $$2n^2: 2, 8, 18, 32, ...$$

Correct answer: $$-2n+3 : 1, -1, -3, -5, ...$$

$$n+1 : 2, 3, 4, 5, ...$$

$$3n-2: 1, 4, 7, 10, ...$$

Q3.
What is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts 3, 3, 5, 9, 15, ...?

$$n^2 + 2$$

$$2n^2 + 1$$

Correct answer: $$n^2 -3n + 5$$

$$n^2 +3n-2$$

$$2n^2 -6n +7$$

Q4.
The $$n^\text{th}$$ term rule for the quadratic sequence 4, 9, 16, 25, ... is $$n^2 + $$ $$n + 1$$

Correct Answer: 2

Q5.
What is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts -3, 1, 11, 27, 49, ... ?

$$n^2 +n -5$$

Correct answer: $$3n^2 -5n -1$$

$$6n^2 -14n +5$$

$$12n^2 -32n+17$$

Q6.
What is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts -1, -6, -15, -28, -45, ... ?

Correct answer: $$-2n^2 + n$$

$$-2n^2 - n + 2$$

$$-4n^2 - n - 2$$

$$-4n^2+ 7n - 4$$

To help you plan your 11 maths lesson on: Quadratic sequences, download all teaching resources for free and adapt to suit your pupils' needs...

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Quadratic sequences

Lesson slides

Lesson details

Learning outcome

Key learning points

Keywords

Common misconception

Teacher tip

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.Which of these could be the first 5 terms of a quadratic sequence?

Q2.Match the first 5 terms of these quadratic sequences to their common second differences.

Q3.Which is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts: 3, 6, 11, 18, 27, ...?

Q4.The quadratic sequence which starts: 13, 16, 21, 28, ... has $$n^\text{th}$$ term rule $$n^2 +$$

Q5.What is the $$n^\text{th}$$ term of the linear sequence which starts -1, -2, -3, -4, ... ?

Q6.What is the $$n^\text{th}$$ term of the linear sequence which starts -1, 2, 5, 8, ... ?

Assessment exit quiz

6 Questions

Q1.Which two sequences from the options below could be added to make the quadratic sequence which starts: 3, 7, 13, 21, ...?

Q2.Which two sequences from the options below could be added to make the quadratic sequence which starts: 3, 7, 15, 27, ...?

Q3.What is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts 3, 3, 5, 9, 15, ...?

Q4.The $$n^\text{th}$$ term rule for the quadratic sequence 4, 9, 16, 25, ... is $$n^2 + $$ $$n + 1$$

Q5.What is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts -3, 1, 11, 27, 49, ... ?

Q6.What is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts -1, -6, -15, -28, -45, ... ?

How to plan a lesson using our resources

Q1.
Which of these could be the first 5 terms of a quadratic sequence?

Q2.
Match the first 5 terms of these quadratic sequences to their common second differences.

Q3.
Which is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts: 3, 6, 11, 18, 27, ...?

Q4.
The quadratic sequence which starts: 13, 16, 21, 28, ... has $$n^\text{th}$$ term rule $$n^2 +$$

Q5.
What is the $$n^\text{th}$$ term of the linear sequence which starts -1, -2, -3, -4, ... ?

Q6.
What is the $$n^\text{th}$$ term of the linear sequence which starts -1, 2, 5, 8, ... ?

Q1.
Which two sequences from the options below could be added to make the quadratic sequence which starts: 3, 7, 13, 21, ...?

Q2.
Which two sequences from the options below could be added to make the quadratic sequence which starts: 3, 7, 15, 27, ...?

Q3.
What is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts 3, 3, 5, 9, 15, ...?

Q4.
The $$n^\text{th}$$ term rule for the quadratic sequence 4, 9, 16, 25, ... is $$n^2 + $$ $$n + 1$$

Q5.
What is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts -3, 1, 11, 27, 49, ... ?

Q6.
What is the correct $$n^\text{th}$$ term rule for the quadratic sequence which starts -1, -6, -15, -28, -45, ... ?