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4PM1/01R Jan 2020, Question 7: Arithmetic Progressions in Problems

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 11 marks

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The question, in short

Arithmetic series P with nth term uₙ; given u₅ = 4x + 6 and u₈ = 7x + 3, work with d, a and x, then solve an equation linking S₍ₙ₊₁₎ and uₙ.

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

Part (a) · 4 marks

Show that d = x − 1 and find the value of a.

  1. 1a + 4d = 4x + 6 and a + 7d = 7x + 3
  2. 2Subtract: 3d = 3x − 3, so d = x − 1
  3. 3a = 4x + 6 − 4(x − 1) = 10

Answerd = x − 1 shown; a = 10

Part (b) · 2 marks

Given u₉ = 42, find the value of x.

  1. 1u₉ = a + 8d = 10 + 8(x − 1) = 42
  2. 28(x − 1) = 32, so x = 5

Answerx = 5

Part (c) · 5 marks

Find n such that S₍ₙ₊₁₎ = 12uₙ + 18.

  1. 1d = 5 − 1 = 4
  2. 2S₍ₙ₊₁₎ = ((n + 1)/2)(20 + 4n) = (n + 1)(10 + 2n)
  3. 312uₙ + 18 = 12(10 + 4(n − 1)) + 18 = 48n + 90
  4. 42n² + 12n + 10 = 48n + 90, so n² − 18n − 40 = 0
  5. 5(n − 20)(n + 2) = 0, so n = 20

Answern = 20

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