4PM1/02R Jan 2023, Question 1: The Binomial Expansion
Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 6 marks
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The question, in short
Expand a binomial power (1 + x/4)⁸ as a series in ascending powers of x up to the x³ term, then use it to approximate (1.035)⁸ to 4 decimal places.
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Part (a) · 3 marks
Write the binomial expansion of (1 + x/4)⁸ in ascending powers of x, stopping at the x³ term, with each coefficient simplified.
- 1(1 + x/4)⁸ = 1 + 8(x/4) + (8 × 7/2!)(x/4)² + (8 × 7 × 6/3!)(x/4)³ + …
- 2= 1 + 2x + 28(x²/16) + 56(x³/64) + …
- 3= 1 + 2x + (7/4)x² + (7/8)x³
Answer1 + 2x + (7/4)x² + (7/8)x³
Part (b) · 3 marks
Choose a value of x, substitute it into the expansion, and hence estimate (1.035)⁸ correct to 4 decimal places.
- 11 + x/4 = 1.035 → x = 0.14
- 2(1.035)⁸ ≈ 1 + 2(0.14) + 1.75(0.14)² + 0.875(0.14)³
- 3= 1.3167 (4 d.p.)
Answer1.3167
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