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4PM1/01R Jan 2022, Question 5: The Binomial Expansion

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 8 marks

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The question, in short

For f(x) = (1 + ax)ⁿ with n > 2, expand up to the x³ term, then use given x and x² coefficients to find n and a, and finally get the x³ coefficient.

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

Part (a) · 2 marks

Give the binomial expansion of (1 + ax)ⁿ up to and including the x³ term, writing each coefficient using n and a.

  1. 1(1 + ax)ⁿ = 1 + n(ax) + n(n − 1)/2! (ax)² + n(n − 1)(n − 2)/3! (ax)³ + …

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

Part (b) · 4 marks

Given that the x coefficient is 8 and the x² coefficient is 30, determine the values of n and a.

  1. 1an = 8, so a = 8/n
  2. 2n(n − 1)/2 × a² = 30
  3. 3n(n − 1)/2 × 64/n² = 30
  4. 432(n − 1) = 30n
  5. 5n = 16, a = 8/16 = 1/2

Answern = 16, a = 1/2

Part (c) · 2 marks

Using your values, work out the numerical coefficient of x³ in the expansion.

  1. 1Coefficient of x³ = n(n − 1)(n − 2)/6 × a³
  2. 2= (16 × 15 × 14)/6 × (1/2)³
  3. 3= 560 × 1/8 = 70

Answer70

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