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Cambridge IGCSE0607

Sequences

International Mathematics 0607 Chapter Notes

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Sequences
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In this chapter

  1. 1Continuing a sequence and the term-to-term rule
  2. 2Square, cube and triangular numbers
  3. 3The nth term of a linear sequence
  4. 4Decreasing linear sequences
  5. 5Using differences to continue a sequence
  6. 6The nth term of a quadratic sequence
  7. 7Cubic sequences
  8. 8Exponential sequences

1. Continuing a sequence and the term-to-term rule

A sequence is a list of numbers that follows a pattern. To continue it, find what happens from one term to the next.

The term-to-term rule says how to get the next term from the one before. Write it in words, such as add 6. Do not put n in it.

Subscript notation is used for terms: Tn is the nth term of sequence T.

T₁, T₂, T₃, …, Tn

Examiner insight

“Some candidates did not know how to express the rule as they included an n in their description.” “Some identified the differences between terms, but then gave the next two differences, not the next two terms, as their final answer.”

Worked example 12 marks

These are the first 5 terms of a sequence.

7 13 19 25 31

(i) Write down the next term of this sequence.

(ii) Write down the rule for finding the next term of this sequence.

  1. 1

    each term is 6 more than the one before: 31 + 6 = 37 [1]

  2. 2

    rule: add 6 (or +6) [1]

2. Square, cube and triangular numbers

Some sequences are named. The square numbers are 1, 4, 9, 16, 25, … and the cube numbers are 1, 8, 27, 64, 125, …

The triangular numbers are 1, 3, 6, 10, 15, … Here the gaps grow by one each time: 2, 3, 4, 5, …

Different sequences can be related. The sequence with nth term n² + n is 2, 6, 12, … Each term is double a triangular number.

n²: 1, 4, 9, 16

n³: 1, 8, 27, 64

1/2n(n+1): 1, 3, 6, 10

Examiner insight

“Candidates need to be able to recognise sequences such as squares, cubes and triangle numbers as these are often found in investigations.”

Worked example 12 marks

The nth term of a sequence is n² + n.

Find the first 3 terms of the sequence.

  1. 1

    put n = 1, n = 2 and n = 3 into n² + n [1]

  2. 2

    1 + 1 = 2, 4 + 2 = 6, 9 + 3 = 12, so 2, 6, 12 [1]

3. The nth term of a linear sequence

In a linear sequence the terms go up (or down) by the same common difference d each time. The nth term is dn plus a constant.

Find d. Write dn. Then compare d × 1 with the first term and add the constant that makes up the difference.

Check the answer by putting in n = 1 and n = 2.

nth term = dn + c

nth term = a + (n-1)d

Examiner insight

“Some candidates used the formula, nth term = a + (n – 1)d, and had problems substituting for a and d or during rearrangement.” “Successful candidates used the difference of 5 to arrive at 5n and then subtracted 5 from 6 to reach +1. The most common wrong answer was n + 5.”

Worked example 12 marks

These are the first four terms of a sequence.

1.5 4 6.5 9

Find the nth term of the sequence.

  1. 1

    common difference = 4 - 1.5 = 2.5, so the nth term starts 2.5n [1]

  2. 2

    2.5 × 1 = 2.5 but the first term is 1.5, so subtract 1: 2.5n - 1 [1]

4. Decreasing linear sequences

When a sequence goes down, the common difference is negative. The n term then has a negative number in front of it.

The constant is still found by comparing d × 1 with the first term.

The answer can be written with the constant first, such as 17 - 3n.

d < 0

nth term = c + dn

Examiner insight

“This was more complex than some other sequences as it is descending.” “Some candidates wrote 54n – 3 instead of 54 – 3n and others just wrote n – 3 for their answer.”

Worked example 13 marks

These are the first 5 terms of a sequence.

14 11 8 5 2

(i) Find the next term.

(ii) Find the nth term.

  1. 1

    next term = 2 - 3 = -1 [1]

  2. 2

    common difference is -3, so the nth term starts -3n [1]

  3. 3

    -3 × 1 = -3, add 17 to reach 14: 17 - 3n [1]

5. Using differences to continue a sequence

When the gaps between terms are not equal, write the first differences under the terms.

If the first differences change by the same amount each time, that change is the second difference. Continue the first differences, then add them on to find the next terms.

A constant second difference means the sequence is quadratic.

second difference constant ⇒ quadratic

Examiner insight

“Most candidates found the next two terms correctly. A few candidates made arithmetical errors but were able to find one term correctly.”

Worked example 12 marks

These are the first 5 terms of a sequence.

17 17 15 11 5

Find the next two terms.

  1. 1

    first differences are 0, -2, -4, -6: the second difference is -2, so the next differences are -8 and -10 [1]

  2. 2

    5 - 8 = -3 and -3 - 10 = -13, so -3 and -13 [1]

6. The nth term of a quadratic sequence

A quadratic sequence has an nth term an² + bn + c. The second difference is 2a, so a is half the second difference.

Subtract an² from each term. What is left is a linear sequence. Find its nth term and add it on.

This is the difference method. Always show the second row of differences.

an² + bn + c

a = 1/2 × second difference

Examiner insight

“Although there were many correct answers for the nth term, those candidates who could not get the correct answer but realised that it must be a quadratic function scored one of the two marks available. Some candidates earned the method mark by showing a second row of differences as 4.” “Some had realised the second difference was 2 and gave answers of n + 2 or in the form 2n + k.”

Worked example 13 marks

Find the nth term of this sequence.

2 5 10 17 26

  1. 1

    first differences are 3, 5, 7, 9 and second differences are 2, 2, 2, so a = 1 and the nth term starts n² [1]

  2. 2

    n² gives 1, 4, 9, 16, 25; the terms minus these are 1, 1, 1, 1, 1 [1]

  3. 3

    nth term = n² + 1 [1]

7. Cubic sequences

In a cubic sequence the first and second differences both change. The third differences are the same each time.

The nth term is an³ + bn² + cn + d. The third difference is 6a. Subtract an³ from each term and look at what is left.

Using a difference method to find a cubic nth term is Extended only.

an³ + bn² + cn + d

third difference = 6a

Examiner insight

“This cubic sequence was more challenging. Many candidates found the correct nth term while others were able to earn a method mark for finding the correct third differences of 6 without realising that this would then lead to a cubic sequence.”

Worked example 12 marks

These are the first 5 terms of another sequence.

1 15 53 127 249

Find the nth term of this sequence.

  1. 1

    first differences 14, 38, 74, 122

  2. 2

    second differences 24, 36, 48; third differences 12, 12

  3. 3

    third difference is 12 = 6a, so a = 2 and the nth term starts 2n³ [1]

  4. 4

    2n³ gives 2, 16, 54, 128, 250; each term is 1 less, so 2n³ - 1 [1]

8. Exponential sequences

In an exponential sequence each term is the one before multiplied by the same number, the common ratio. The ratio can be a fraction or negative.

A negative ratio makes the terms alternate in sign. Exponential sequences are Extended only.

nth term = a × rⁿ⁻¹

Examiner insight

“The nth term was often a negative expression rather than an alternating sign expression.” “Many candidates were able to score at least one mark by realising that the nth term must include a power of 2.”

Worked example 13 marks

These are the first 5 terms of a sequence.

80 -40 20 -10 5

(a) Write down the next term.

(b) Find the nth term.

  1. 1

    each term is the one before × (-1/2), so next term = 5 × (-1/2) = -2.5 [1]

  2. 2

    first term 80 and common ratio -1/2 [1]

  3. 3

    nth term = 80 × (-1/2)ⁿ⁻¹ [1]

Recap

  • The term-to-term rule is written in words, with no n: add 6.
  • Square, cube and triangular numbers: 1, 4, 9, 16; 1, 8, 27, 64; 1, 3, 6, 10.
  • Linear: nth term = dn + c. A decreasing sequence has a negative d: 17 - 3n.
  • Quadratic: constant second difference; a is half of it.
  • Cubic: constant third difference; a is one sixth of it.
  • Exponential: multiply by a common ratio; nth term = a × rⁿ⁻¹.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. The first four terms of a sequence are 100, 91, 82, 73. (a) Write down the next two terms. (b) Write down the term-to-term rule.3 marks
  2. Do not use a calculator. (a) Write down the next term of the sequence 1, 8, 27, 64 and name the sequence. (b) Write down the next two terms of the sequence 1, 3, 6, 10, 15 and name the sequence.4 marks
  3. Do not use a calculator. The nth term of a sequence is n² - 3. Work out the first three terms and the 10th term.3 marks
  4. The first four terms of a sequence are 40, 35, 30, 25. (a) Find the nth term. (b) Which term of the sequence is equal to -10?4 marks
  5. The nth term of a sequence is 8n - 1. Is 120 a term of this sequence? Show how you decide.3 marks
  6. Do not use a calculator. Find the nth term of the sequence 5, 11, 21, 35, 53.3 marks
  7. The first four terms of a sequence are 81, -27, 9, -3. (a) Write down the next term. (b) Find the nth term.3 marks
  8. A scientist records the readings 9, 2, -17, -54, -115 in an experiment. (a) Show that the readings form a cubic sequence. (b) Find the nth term. (c) Work out the 10th reading.5 marks
  9. A gardener lays paving slabs in rings around a pond. After 1, 2, 3, 4 and 5 rings the total numbers of slabs used are 4, 11, 22, 37, 56. (a) Find the nth term of this sequence. (b) The gardener has 220 slabs. Work out the greatest number of complete rings that can be laid and the number of slabs left over.6 marks
  10. Do not use a calculator. A computer file is copied in rounds. After rounds 1 to 5 the total numbers of copies are 3, 7, 15, 31, 63. (a) Use differences to show that this sequence is not linear, quadratic or cubic. (b) Find the nth term of the sequence of first differences. (c) Find the nth term of the sequence of totals. (d) Find the first round after which the total is more than 1000.8 marks

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