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4PM1/01 Nov 2023, Question 7: Geometric Progressions and Sum to Infinity

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 9 marks

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The question, in short

Geometric series with first term 16 and third term 2704/625: find possible common ratios, then with r > 0 find the infinite sum and the least n for which the sum to n terms exceeds 33.

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

Part (a) · 2 marks

Find the two possible values of the common ratio r.

  1. 1ar² = 2704/625 with a = 16
  2. 2r² = 2704/(625 × 16) = 169/625
  3. 3r = ±13/25

Answerr = 13/25 or r = −13/25

Part (b) · 2 marks

Given r > 0, find the sum to infinity of G.

  1. 1S∞ = 16/(1 − 13/25)
  2. 2= 16/(12/25) = 100/3

Answer100/3

Part (c) · 5 marks

Using logarithms, find the least n for which the sum to n terms is greater than 33, showing working.

  1. 116(1 − (13/25)ⁿ)/(1 − 13/25) > 33
  2. 21 − (13/25)ⁿ > 0.99
  3. 3(13/25)ⁿ < 0.01
  4. 4n lg(13/25) < lg 0.01
  5. 5lg(13/25) < 0, so n > lg 0.01/lg(13/25) = 7.04…
  6. 6Least n = 8

Answern = 8

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