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4PM1/01R Jun 2024, Question 2: Binomial Series for Any Rational Power

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

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The question, in short

Binomial expansion 1 − (1/3)x + (5/36)x² + … equals (1 + Ax)ⁿ with rational A and n; find A and n, then the x³ coefficient as −p/q with p prime.

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Part (a) · 6 marks

Use the given first three terms to find the rational values of A and n in (1 + Ax)ⁿ.

  1. 1(1 + Ax)ⁿ = 1 + nAx + n(n − 1)(Ax)²/2 + …
  2. 2nA = −1/3 and n(n − 1)A²/2 = 5/36
  3. 3A = −1/(3n), so n(n − 1)/(9n²) × 1/2 = 5/36
  4. 4(n − 1)/(18n) = 5/36 gives n = −2/3
  5. 5A = −1/(3 × (−2/3)) = 1/2

AnswerA = 1/2, n = −2/3

Part (b) · 2 marks

Using your A and n, find the coefficient of x³ and express it as −p/q, p prime.

  1. 1Coefficient of x³ = n(n − 1)(n − 2)A³/3!
  2. 2= (−2/3)(−5/3)(−8/3)(1/2)³/6
  3. 3= −5/81

Answer−5/81

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