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4PM1/02 Nov 2023, Question 8: Sigma Notation

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 10 marks

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The question, in short

Use an arithmetic series with first four terms summing to 42 and fifth term 23 to derive a summation form and solve for n.

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

Part (a) · 6 marks

Show the sum to n terms equals Σ(Pr − Q) with P and Q prime, using the given 42 and 23.

  1. 1S₄ = (4/2)(2a + 3d) = 42, so 4a + 6d = 42
  2. 2a + 4d = 23
  3. 3Solve: a = 3, d = 5
  4. 4nth term = 3 + (n − 1)5 = 5n − 2
  5. 5Sₙ = Σ from r = 1 to n of (5r − 2)

AnswerSₙ = Σ from r = 1 to n of (5r − 2) (P = 5, Q = 2)

Part (b) · 4 marks

Given S₂ₙ − 3Uₙ = 1062, find the value of n.

  1. 1S₂ₙ = (2n/2)(2(3) + (2n − 1)5) = n(10n + 1)
  2. 23Uₙ = 3(5n − 2) = 15n − 6
  3. 310n² + n − 15n + 6 = 1062
  4. 410n² − 14n − 1056 = 0, (5n + 48)(2n − 22) = 0
  5. 5n = 11 (n = −48/5 rejected)

Answern = 11

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