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WMA12/01 Oct 2022, Question 8: Geometric Progressions and Sum to Infinity

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 7 marks

The question, in short

A geometric series with first term a and ratio r has infinite sum 3a; prove the ratio equals 2/3, then use u₂ − u₄ = 16 to calculate S₁₀ to one decimal place.

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

Part (a) · 2 marks

Using the infinite sum formula set equal to 3a, derive that the common ratio is 2/3.

  1. 1S∞ = a/(1 − r) = 3a
  2. 2Divide by a: 1 = 3(1 − r) = 3 − 3r
  3. 33r = 2, so r = 2/3

AnswerShown: a/(1 − r) = 3a ⇒ 1 = 3(1 − r) ⇒ r = 2/3

Part (b) · 5 marks

Given the difference between the second and fourth terms is 16, work out the sum of the first ten terms, rounded to one decimal place.

  1. 1u₂ − u₄ = ar − ar³ = 16
  2. 2a(2/3 − 8/27) = 16, so a × 10/27 = 16
  3. 3a = 43.2
  4. 4S₁₀ = 43.2(1 − (2/3)¹⁰)/(1 − 2/3)
  5. 5= 129.6 × (1 − 0.01734…) = 127.35…
  6. 6S₁₀ = 127.4 (1 d.p.)

AnswerS₁₀ = 127.4

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Also searched as: WMA12 October 2022 · WMA12/01 · wma12 oct22 · IAS Maths Pure Mathematics 2 October 2022 · question 8.
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