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WMA12/01 Jun 2021, Question 4: The Binomial Expansion

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Expand (2 + x/8)^13 in increasing powers of x as far as x³ with simplified coefficients, then use it to estimate 2.0125^13 and judge the estimate's accuracy.

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

Part (a) · 4 marks

Write the binomial expansion of (2 + x/8)^13 in ascending powers of x through the x³ term, simplifying every coefficient.

  1. 1First term: 2¹³ = 8192
  2. 2x term: 13 × 2¹² × (x/8) = 6656x
  3. 3x² term: 78 × 2¹¹ × (x/8)² = 2496x²
  4. 4x³ term: 286 × 2¹⁰ × (x/8)³ = 572x³

Answer8192 + 6656x + 2496x² + 572x³

Part (b) · 3 marks

Using the expansion from part (a), approximate 2.0125^13 and round the result to 3 decimal places.

  1. 12 + x/8 = 2.0125 gives x = 0.1
  2. 28192 + 6656(0.1) + 2496(0.01) + 572(0.001)
  3. 3= 8192 + 665.6 + 24.96 + 0.572 = 8883.132

Answer8883.132

Part (c) · 1 mark

Without computing 2.0125^13 directly, decide whether the part (b) value is too high or too low and justify it.

  1. 1With x = 0.1 every term of the expansion is positive
  2. 2Missing terms are positive, so the truncated sum is too small: an underestimate

AnswerUnderestimate, because every term in the expansion is positive, so the terms left out would add to the value.

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