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0580/33 May/Jun 2022, Question 4: Pattern Sequences and Special Sequences

Cambridge IGCSE Mathematics 0580 · Paper 33 (Core) · Calculator · 12 marks

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The question, in short

For a regular polygon shown in a diagram, name it, verify its interior angle is 135°, then analyse a sequence of joined copies counting lines to build a table, rule, formula and solve for a diagram number.

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

Part (ai) · 1 mark

Give the mathematical name of the regular polygon shown.

  1. 1The polygon has 8 sides: octagon

AnswerOctagon

Part (aii) · 2 marks

Demonstrate, with working, that this polygon's interior angle equals 135°.

  1. 1Exterior angle = 360 ÷ 8 = 45°
  2. 2Interior angle = 180 − 45 = 135°

Answer180 − (360 ÷ 8) or ((8 − 2) × 180)/8

Part (bi) · 3 marks

Fill in the missing line counts for diagrams 4 and 5 in the table.

  1. 1Each new octagon adds 7 lines: 8, 15, 22, 29, 36
  2. 2Diagram 3: 22, Diagram 4: 29, Diagram 5: 36

Answer22, 29, 36

Part (bii) · 1 mark

State how the line count changes from one diagram to the next.

  1. 115 − 8 = 7, 22 − 15 = 7
  2. 2Add 7

AnswerAdd 7

Part (biii) · 1 mark

Calculate how many lines diagram 9 contains.

  1. 1Diagrams 1, 2, 3 have 8, 15, 22 lines: add 7 each time
  2. 2Diagram 9 = 8 + 8 × 7
  3. 3= 64 lines

Answer64

Part (biv) · 2 marks

Derive a formula using n for the line count of diagram n.

  1. 1Difference 7, so 7n + c
  2. 2n = 1: 7 + c = 8, so c = 1
  3. 3Number of lines = 7n + 1

Answer7n + 1

Part (bv) · 2 marks

Given diagram k contains 113 lines, determine k.

  1. 17k + 1 = 113
  2. 27k = 112
  3. 3k = 16

Answer16

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