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4PM1/01 Jun 2021, Question 5: Binomial Series for Any Rational Power

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 10 marks

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The question, in short

Expand (1 + ax)ⁿ up to the x³ term with coefficients in a and n, then use given coefficient conditions to find a and n and the x³ coefficient.

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

Part (a) · 2 marks

Expand (1 + ax)ⁿ in ascending powers of x through x³, giving each x coefficient in terms of a and n.

  1. 1(1 + ax)ⁿ = 1 + n(ax) + (n(n − 1)/2!)(ax)² + (n(n − 1)(n − 2)/3!)(ax)³ + …
  2. 2= 1 + anx + (n(n − 1)/2)a²x² + (n(n − 1)(n − 2)/6)a³x³

Answer1 + anx + (n(n − 1)/2)a²x² + (n(n − 1)(n − 2)/6)a³x³

Part (b) · 6 marks

Given the x coefficient is 15 and the x² and x³ coefficients are equal, find a and n.

  1. 1an = 15
  2. 2(n(n − 1)/2)a² = (n(n − 1)(n − 2)/6)a³
  3. 3Divide by n(n − 1)a²/2: 3 = (n − 2)a
  4. 43 = an − 2a = 15 − 2a, so a = 6
  5. 5n = 15/6 = 2.5

Answera = 6, n = 2.5

Part (c) · 2 marks

Find the coefficient of x³.

  1. 1Coefficient of x³ = (n(n − 1)(n − 2)/6)a³
  2. 2= (2.5 × 1.5 × 0.5 × 6³)/6
  3. 3= 67.5

Answer67.5

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