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0606/13 Oct/Nov 2025, Question 12: The Binomial Expansion

Cambridge IGCSE Additional Mathematics 0606 · Paper 13 (Non-calculator) · Non-calculator · 5 marks

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The question, in short

Use the binomial expansion of (2 + x)⁵⁹ to state a general coefficient and then find which r makes consecutive coefficients equal.

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

Part (a) · 1 mark

State the coefficient of x^r in the expansion of (2 + x)⁵⁹.

  1. 1General term: ⁵⁹Cᵣ × 2^(59−r) × x^r
  2. 2Coefficient of x^r = ⁵⁹Cᵣ × 2^(59−r) = 59!/((59 − r)! r!) × 2^(59−r)

Answer⁵⁹Cᵣ × 2^(59−r)

Part (b) · 4 marks

Find the value of r where the coefficients of x^r and x^(r+1) are equal.

  1. 159!/((59 − r)! r!) × 2^(59−r) = 59!/((58 − r)! (r + 1)!) × 2^(58−r)
  2. 2Divide both sides by 59! × 2^(58−r)/((58 − r)! r!)
  3. 32/(59 − r) = 1/(r + 1)
  4. 42r + 2 = 59 − r
  5. 5r = 19

Answerr = 19

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Also searched as: 0606/13/O/N/25 · 0606_w25_qp_13 · 0606 w25 13 · question 12.
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