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Number sequences · 4 marks

Can you describe the pattern in each of the above sequences? Can you predict the next few numbers in these sequences?

  1. (i) 1, 2, 3, 4, 5, 6, … (Natural Numbers)
  2. (ii) 1, 3, 5, 7, 9, 11, … (Odd Numbers)
  3. (iii) 1, 3, 6, 10, 15, 21, … (Triangular Numbers)
  4. (iv) 1, 4, 9, 16, 25, 36, … (Square Numbers)
Answer: (i) Add 1 each time: 7, 8, 9, … (ii) Add 2 each time: 13, 15, 17, … (iii) Add 2, then 3, then 4, …: 28, 36, 45, … (iv) Add 3, 5, 7, … (the squares 1², 2², 3², …): 49, 64, 81, …

Step-by-step solution

Idea: To spot the pattern, look at the difference between each term and the one before it. If the difference is fixed, keep adding it. If the differences themselves follow a pattern, continue that pattern.

(i) 1, 2, 3, 4, 5, 6, … (Natural Numbers)

  1. Differences: 2 − 1 = 1, 3 − 2 = 1, … Every term is 1 more than the term before it.½ mark
  2. Next numbers: 6 + 1 = 7, then 8, 9, 10.½ mark
Add 1 each time; next: 7, 8, 9, 10

(ii) 1, 3, 5, 7, 9, 11, … (Odd Numbers)

  1. Differences: 3 − 1 = 2, 5 − 3 = 2, … Every term is 2 more than the term before it.½ mark
  2. Next numbers: 11 + 2 = 13, then 15, 17, 19.½ mark
Add 2 each time; next: 13, 15, 17, 19

(iii) 1, 3, 6, 10, 15, 21, … (Triangular Numbers)

  1. Differences: 2, 3, 4, 5, 6. The amount added goes up by 1 each time. So the nth term is 1 + 2 + 3 + … + n (for example, 10 = 1 + 2 + 3 + 4).½ mark
  2. Next differences are 7, 8, 9: 21 + 7 = 28, 28 + 8 = 36, 36 + 9 = 45.½ mark
Add 2, 3, 4, 5, …; next: 28, 36, 45

(iv) 1, 4, 9, 16, 25, 36, … (Square Numbers)

  1. Each term is a number multiplied by itself: 1 × 1, 2 × 2, 3 × 3, … The differences are the odd numbers 3, 5, 7, 9, 11.½ mark
  2. Next numbers: 7 × 7 = 49, 8 × 8 = 64, 9 × 9 = 81 (check: 36 + 13 = 49, 49 + 15 = 64, 64 + 17 = 81).½ mark
Squares 1², 2², 3², …; next: 49, 64, 81
Natural numbers: add 1 (next 7, 8, 9). Odd numbers: add 2 (next 13, 15, 17). Triangular numbers: add 2, 3, 4, … (next 28, 36, 45). Square numbers: n × n, differences 3, 5, 7, … (next 49, 64, 81).

Check: Triangular: the 7th term should be 1 + 2 + … + 7 = 28 ✓. Square: the 8th term should be 8 × 8 = 64 ✓.

Answer to write in the exam

(i)

Difference = 1 (each term = previous term + 1)

∴ Next numbers: 7, 8, 9, 10

(ii)

Difference = 2 (each term = previous term + 2)

∴ Next numbers: 13, 15, 17, 19

(iii)

Differences: 2, 3, 4, 5, 6, …

nth term = 1 + 2 + … + n

21 + 7 = 28, 28 + 8 = 36, 36 + 9 = 45

∴ Next numbers: 28, 36, 45

(iv)

Terms: 12, 22, 32, … (differences 3, 5, 7, …)

72 = 49, 82 = 64, 92 = 81

∴ Next numbers: 49, 64, 81

Common mistakes that cost marks

  • Adding a fixed 5 to 21 for the triangular numbers (getting 26). The amount added grows by 1 each time, so the next difference is 7, not 5 or 6.
  • Treating the square numbers as ‘add 3 each time’ because 4 − 1 = 3. Check more than one difference: they are 3, 5, 7, …, so the next square after 36 is 49.
  • Describing a pattern only from the first two terms. Always check the rule on every term you are given.

How this can come in the exam

MCQ (1 mark)

What is the 8th number in the sequence of square numbers 1, 4, 9, 16, …?

  1. 36
  2. 49
  3. 64
  4. 81
Show answer

(C) 64
The 8th square number is 8 × 8 = 64.

Short answer (2 marks)

Describe the pattern in 1, 3, 6, 10, 15, … and find the 9th term.

Show answerThe differences are 2, 3, 4, 5, …, so the nth term is 1 + 2 + … + n (1 mark). The terms continue 21, 28, 36, 45, so the 9th term is 45 (1 mark).

Try one yourself

Describe the pattern and write the next three numbers: 2, 6, 12, 20, 30, …

Show answer

Differences are 4, 6, 8, 10 (going up by 2). Next differences 12, 14, 16 give 42, 56, 72. (Each term is n × (n + 1).)

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