Lemmara logoLemmaraWorked solution

4024/22 May/Jun 2024, Question 4: Pattern Sequences and Special Sequences

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 9 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Using triangular-grid patterns, complete a table, derive expressions for triangle counts in terms of n, and work with a linear sequence having 3rd term 34 and 8th term 14.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (ai) · 1 mark

Sketch the fifth pattern of the sequence on the provided triangular dot grid.

  1. 1Pattern 5 is a large triangle with 5 rows: 25 small triangles
  2. 2Shade the 10 upside-down small triangles grey

AnswerPattern 5 drawn: a large triangle of 25 small triangles with the 10 inverted small triangles shaded grey

Part (aii) · 2 marks

Fill in the missing entries for patterns 4, 5 and 6 in the totals, grey and white triangle rows.

  1. 1Total = n²: 16, 25, 36
  2. 2Grey goes 0, 1, 3, 6, 10, 15 (add 1, 2, 3, 4, 5)
  3. 3White = total − grey: 10, 15, 21

Answer| Pattern (n) | 4 | 5 | 6 | | Total number of triangles | 16 | 25 | 36 | | Number of grey triangles | 6 | 10 | 15 | | Number of white triangles | 10 | 15 | 21 |

Part (aiii) · 1 mark

Give a formula involving n for how many triangles Pattern n contains altogether.

  1. 11, 4, 9, 16 are square numbers: n²

Answern²

Part (aiv) · 2 marks

Give a formula involving n for how many white triangles Pattern n contains.

  1. 1White: 1, 3, 6, 10; differences 2, 3, 4; second difference 1
  2. 2Second difference 1, so the n² term is (1/2)n²
  3. 3Take away (1/2)n²: 1/2, 1, 3/2, 2, which is (1/2)n
  4. 4White triangles = (1/2)n² + (1/2)n

Answer(1/2)n² + (1/2)n

Part (bi) · 2 marks

Work out the starting value of the linear sequence whose 3rd term is 34 and 8th term is 14.

  1. 15 differences from 34 to 14: d = (14 − 34)/5 = −4
  2. 2First term = 34 − 2 × (−4) = 42

Answer42

Part (bii) · 1 mark

Determine the value of the earliest term in this sequence that is below zero.

  1. 1nth term = 42 − 4(n − 1) = 46 − 4n
  2. 211th term = 2, 12th term = −2
  3. 3First negative term = −2

Answer−2

Stuck on a step? Read the free lesson: Pattern Sequences and Special Sequences

Practise more pattern sequences and special sequences questions

Real 4024 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on pattern sequences and special sequences

Also searched as: 4024/22/M/J/24 · 4024_s24_qp_22 · 4024 s24 22 · question 4.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.