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WMA12/01 Jan 2023, Question 7: Geometric Progressions and Sum to Infinity

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

The question, in short

Geometric sequence with first term a and ratio r > 0 has 3rd term 20 and 5th term 12.8; show r = 0.8, find a, and find the smallest n where the sum of the first n terms exceeds 156.

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

Part (a) · 1 mark

Show that the common ratio r equals 0.8 using the given 3rd and 5th terms.

  1. 1ar² = 20 and ar⁴ = 12.8
  2. 2r² = 12.8/20 = 0.64
  3. 3r > 0 so r = 0.8

AnswerShown: ar⁴ ÷ ar² = r² = 12.8/20 = 0.64, so r = 0.8

Part (b) · 2 marks

Use r = 0.8 to find the first term a.

  1. 1ar² = 20 ⇒ a = 20 ÷ 0.8²
  2. 2a = 20 ÷ 0.64 = 31.25

Answera = 31.25

Part (c) · 4 marks

Find the smallest n for which the sum of the first n terms is greater than 156, without purely graphical or numerical methods.

  1. 131.25(1 − 0.8ⁿ)/(1 − 0.8) > 156
  2. 2156.25(1 − 0.8ⁿ) > 156 ⇒ 1 − 0.8ⁿ > 0.9984
  3. 30.8ⁿ < 0.0016
  4. 4n > log 0.0016 / log 0.8 = 28.85 (inequality flips: log 0.8 < 0)
  5. 5Smallest n = 29

Answern = 29

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