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0580/33 May/Jun 2023, Question 5: Sum of an Arithmetic Series

Cambridge IGCSE Mathematics 0580 · Paper 33 (Core) · Calculator · 14 marks

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The question, in short

Using a stick pattern shown in three diagrams on a grid, work out later diagrams, a formula for diagram n, and stick totals for sequences of diagrams.

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

Part (a) · 1 mark

Draw the fourth diagram of the stick sequence on the blank grid.

  1. 1Each diagram adds one more house shape
  2. 2Diagram 4: 4 roof pairs (8 sloping sticks), 5 vertical sticks and 4 base sticks, 17 sticks

AnswerCorrect diagram drawn

Part (b) · 2 marks

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

  1. 1The number of sticks goes up by 4 each time
  2. 213 + 4 = 17, 17 + 4 = 21

Answer17, 21

Part (ci) · 2 marks

Write an expression in n giving the stick count for diagram n.

  1. 1Difference 4, so 4n
  2. 2n = 1: 4 × 1 + 1 = 5
  3. 34n + 1

Answer4n + 1

Part (cii) · 2 marks

Determine which diagram number contains 73 sticks.

  1. 14n + 1 = 73
  2. 24n = 72
  3. 3n = 18

Answer18

Part (di) · 1 mark

Demonstrate that the first three diagrams together use 27 sticks.

  1. 15 + 9 + 13 = 27

Answer5 + 9 + 13 = 27

Part (dii) · 2 marks

Verify that 2k² + 3k gives the correct total for the first 3 diagrams.

  1. 1Count the sticks: 5 + 9 + 13 = 27 for the first 3 diagrams.
  2. 2Put k = 3 in the formula: 2 × 3² + 3 × 3
  3. 3= 18 + 9 = 27, the same total, so it is correct: 27

Answer2 × 3² + 3 × 3 leading to 27

Part (diii) · 4 marks

Given 240 sticks after building the first 3 diagrams, find how many remain once all 10 diagrams are built.

  1. 1Total sticks in the first n diagrams = 2n² + 3n
  2. 2First 10 diagrams: 2 × 10² + 3 × 10 = 230 sticks
  3. 3Still needed after the first 3: 230 − 27 = 203
  4. 4Sticks left = 240 − 203 = 37

Answer37

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Also searched as: 0580/33/M/J/23 · 0580_s23_qp_33 · 0580 s23 33 · question 5.
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