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4PM1/02R Jan 2020, Question 7: Solving Exponential and Log Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 13 marks

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The question, in short

Geometric series with u₄ = e^(x+2) and u₇ = e^((2x+1)/2): show the common ratio, find the first term, express p, and find the least integer x.

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

Part (a) · 3 marks

Show that the common ratio r equals e^(−1/2) using the two given terms.

  1. 1u₇/u₄ = r³ = e^((2x+1)/2) / e^(x+2)
  2. 2r³ = e^(x + 1/2 − x − 2) = e^(−3/2)
  3. 3r = e^(−1/2)

Answerr = e^(−1/2) shown

Part (b) · 3 marks

Find the first term a in terms of e and x.

  1. 1u₄ = ar³, so a = e^(x+2)/e^(−3/2)
  2. 2a = e^(x + 2 + 3/2) = e^(x + 7/2)

Answera = e^(x + 7/2)

Part (c) · 3 marks

Given the sum to infinity equals e^p/(e^(1/2) − 1), find p in terms of x.

  1. 1S = a/(1 − r) = e^(x + 7/2)/(1 − e^(−1/2))
  2. 2Multiply top and bottom by e^(1/2): S = e^(x + 4)/(e^(1/2) − 1)
  3. 3p = x + 4

Answerp = x + 4

Part (d) · 4 marks

Given u₁₈ > 1.6 and x is an integer, find the least value of x.

  1. 1u₁₈ = ar¹⁷ = e^(x + 7/2) × e^(−17/2) = e^(x − 5)
  2. 2e^(x − 5) > 1.6
  3. 3x − 5 > ln 1.6, so x > 5.470…
  4. 4Least integer x = 6

Answerx = 6

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