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4PM1/02R Jan 2022, Question 5: The Laws of Logarithms

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 10 marks

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The question, in short

For a geometric series with first term 12 and common ratio 3/8, work out its infinite sum, rewrite the 6th term as a power expression, and derive a log relation for the nth term.

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

Part (a) · 2 marks

Calculate the total of the series as it continues without end.

  1. 1S∞ = a/(1 − r)
  2. 2= 12/(1 − 3/8) = 12/(5/8)
  3. 3= 96/5

Answer96/5

Part (b) · 3 marks

Demonstrate that the sixth term equals 3⁶ ÷ 2¹³.

  1. 1u₆ = ar⁵ = 12 × (3/8)⁵
  2. 212 = 2² × 3, 8 = 2³
  3. 3u₆ = 2² × 3 × 3⁵/2¹⁵
  4. 4= 3⁶/2¹³

Answeru₆ = 3⁶/2¹³ shown

Part (c) · 5 marks

Form an expression for uₙ using n, then prove log₂ uₙ = n log₂ 3 − 3n + 5.

  1. 1uₙ = 12 × (3/8)ⁿ⁻¹
  2. 2log₂ uₙ = log₂ 12 + (n − 1) log₂ (3/8)
  3. 3= log₂ 12 + (n − 1)(log₂ 3 − log₂ 8)
  4. 4= log₂ 3 + 2 + (n − 1)(log₂ 3 − 3)
  5. 5= n log₂ 3 − 3n + 5

Answerlog₂ uₙ = n log₂ 3 − 3n + 5 shown

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