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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

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

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