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Securing understanding of arithmetic sequences

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Lesson details

Learning outcome

I can begin to generalise a sequence.

Key learning points

  1. The n^th term is the generalised way of expressing any term in a sequence.
  2. For an arithmetic sequence, it will have the form d(n-1)+a
  3. The a refers to the first term of the sequence.
  4. The d refers to the common difference between any two consecutive terms.

Keywords

  • Arithmetic/ linear sequence - An arithmetic (or linear) sequence is a sequence where the difference between successive terms is constant

Common misconception

Misinterpreting the values in an expression for the n^th term. For example, given the n^th term $$3n-5$$ pupils may think that this relates to a sequence that is decreasing by 5 each time

Remind students about how the expression for the n^th term relates to multiples of a number (times tables) and then a shift. For example, 4, 9, 14, 19, ... can be seen as the 5 times table shifted down 1.

Teacher tip

This lesson is an opportunity to review and build fluency with finding and using the n^th term of arithmetic sequences and for students who are confident they will enjoy exploring a different way to think about the n^th term through $$d(n-1)+a$$

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).

Lesson video

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Prior knowledge starter quiz

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6 Questions

Q1.
The arithmetic (linear) sequence 17, 21, 25, 29, 33, ... has a(n) _________ of +4.

$$n^{\text{th}}$$ term
$$1^{\text{st}}$$ term
Correct answer: common difference
multiplier

Q2.
Which of these sequences are arithmetic?

Correct answer: 2, 6, 10, 14, ...
Correct answer: 2, 7, 12, 17, ...
2, 8, 15, 23, ...
2, 4, 8, 16, ...
Correct answer: 2, -1, -4, -7, ...

Q3.
Match the $$n^{\text{th}}$$ term to the first five terms of each sequence.

Correct Answer:$$3n+2$$,$$5, 8, 11, 14, 17, ...$$

$$5, 8, 11, 14, 17, ...$$

Correct Answer:$$2n+3$$,$$5, 7, 9, 11, 13, ...$$

$$5, 7, 9, 11, 13, ...$$

Correct Answer:$$6n-1$$,$$5, 11, 17, 23, 29, ...$$

$$5, 11, 17, 23, 29, ...$$

Correct Answer:$$6-n$$,$$5, 4, 3, 2, 1, ...$$

$$5, 4, 3, 2, 1, ...$$

Correct Answer:$$5+n$$,$$6, 7, 8, 9, 10, ...$$

$$6, 7, 8, 9, 10, ...$$

Correct Answer:$$1-6n$$,$$-5, -11, -17, -23, -29, ...$$

$$-5, -11, -17, -23, -29, ...$$

Q4.
The solution to the two-step equation $$8x-13=83$$ is $$x=$$

Correct Answer: 12, x=12, twelve

Q5.
Which is true of the solution to the equation $$3x-10=3$$

It is a positive integer.
It is a negative integer.
Correct answer: It is non-integer.
It has no solution.

Q6.
$$8, 13, 18, 23, 28, ...$$ are the first five terms of which sequence?

$$8n+5$$
$$5n+8$$
Correct answer: $$5n+3$$
$$5n-3$$

Assessment exit quiz

Download exit quiz questions (PDF)

6 Questions

Q1.
In the $$n^{\text{th}}$$ term expression $$9n-7$$, you call $$9$$ the __________ of $$n$$.

$$1^{st}$$ term
Correct answer: coefficient
equation
expression

Q2.
What is the common difference in the arithmetic sequence $$6n-3$$?

-6
-3
3
Correct answer: 6

Q3.
Match each sequence to its $$n^{\text{th}}$$ term.

Correct Answer:$$3n+9$$,$$12, 15, 18, 21, 24, ...$$

$$12, 15, 18, 21, 24, ...$$

Correct Answer:$$9n+3$$,$$12, 21, 30, 39, 48, ...$$

$$12, 21, 30, 39, 48, ...$$

Correct Answer:$$9n-3$$,$$6, 15, 24, 33, 42, ...$$

$$6, 15, 24, 33, 42, ...$$

Correct Answer:$$3n-9$$,$$-6,-3,0,3,6, ...$$

$$-6,-3,0,3,6, ...$$

Correct Answer:$$9-3n$$,$$6,3,0,-3,-6, ...$$

$$6,3,0,-3,-6, ...$$

Q4.
405 is a term in the arithmetic sequence with the rule $$7n-50$$. What is its position in the sequence? $$n=$$

Correct Answer: 65, sixty-five, n=65, 65th

Q5.
Is $$189$$ a term in the arithmetic sequence $$7n-50$$?

Yes, because $$189$$ is odd and the sequence contains a lot of odd numbers.
Yes, because $$189\div7=27$$
No, because $$189$$ is odd and the sequence is mostly even.
Correct answer: No, because $$7n-50=189$$ has a non-integer solution.

Q6.
Which expressions could be used to generalise the arithmetic sequence 5, 9, 13, 17, 21, ...?

$$5n+4$$
Correct answer: $$4n+1$$
$$5+4n$$
Correct answer: $$5+4(n-1)$$
$$4+5(n-1)$$

To help you plan your 9 maths lesson on: Securing understanding of arithmetic sequences, download all teaching resources for free and adapt to suit your pupils' needs...