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4PM1/01 Jun 2025, Question 6: Binomial Series for Any Rational Power

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

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The question, in short

Rewrite the surd √(4 − x) in the form A(1 − x/B)^(1/2) with integer A and B, expand it in powers of x, then use the expansion to estimate √305/10.

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

Part (a) · 2 marks

Find integers A and B so that √(4 − x) equals A times (1 − x/B) raised to the power 1/2.

  1. 1√(4 − x) = √4 × √(1 − x/4)
  2. 2= 2(1 − x/4)^(1/2)
  3. 3A = 2, B = 4

AnswerA = 2, B = 4

Part (b) · 3 marks

Expand √(4 − x) as a series in increasing powers of x through the x³ term, giving each coefficient as an exact fraction in lowest terms.

  1. 12[1 + (1/2)(−x/4) + (1/2)(−1/2)(−x/4)²/2! + (1/2)(−1/2)(−3/2)(−x/4)³/3!]
  2. 2= 2[1 − x/8 − x²/128 − x³/1024]
  3. 3= 2 − x/4 − x²/64 − x³/512

Answer2 − x/4 − x²/64 − x³/512

Part (c) · 3 marks

Choose a suitable x value, substitute into your series, and estimate √305/10 to 5 decimal places.

  1. 1√305/10 = √3.05 = √(4 − 0.95), so x = 0.95
  2. 22 − 0.95/4 − 0.95²/64 − 0.95³/512
  3. 3= 1.7467238… ≈ 1.74672

Answer1.74672

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