WMA12/01 Jan 2023, Question 3: The Binomial Expansion
Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 7 marks
The question, in short
Expand f(x) = (2 + kx/8)⁷ up to the x³ term in simplest form, then use the x, x² and x³ coefficients forming an arithmetic progression to find k algebraically.
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Part (a) · 4 marks
Write the first four terms of the binomial expansion of (2 + kx/8)⁷ in ascending powers of x, simplifying each term.
- 1(2 + kx/8)⁷ = 2⁷ + ⁷C₁·2⁶(kx/8) + ⁷C₂·2⁵(kx/8)² + ⁷C₃·2⁴(kx/8)³ + …
- 2= 128 + 7·64·(kx/8) + 21·32·(k²x²/64) + 35·16·(k³x³/512) + …
- 3= 128 + 56kx + (21/2)k²x² + (35/32)k³x³ + …
Answer128 + 56kx + (21/2)k²x² + (35/32)k³x³
Part (b) · 3 marks
Given the x, x² and x³ coefficients are consecutive terms of an arithmetic progression, find the possible values of k using algebra.
- 1AP: (35/32)k³ − (21/2)k² = (21/2)k² − 56k
- 2Divide by k (k ≠ 0): (35/32)k² − 21k + 56 = 0
- 3Multiply by 32/7: 5k² − 96k + 256 = 0
- 4(5k − 16)(k − 16) = 0
- 5k = 16 or k = 16/5
Answerk = 16 or k = 16/5
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