4PM1/01 Jun 2022, Question 4: Geometric Progressions and Sum to Infinity
Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 7 marks
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The question, in short
Geometric series G has positive common ratio, first 4 terms sum to 80 and sum to infinity is 81. Show the first 7 terms' sum differs from the infinite sum by 1/27.
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Solution · 7 marks
- 1a(1 − r⁴)/(1 − r) = 80 and a/(1 − r) = 81
- 2Substitute: 81(1 − r⁴) = 80 ⇒ r⁴ = 1/81
- 3r > 0 so r = 1/3, and a = 81(1 − 1/3) = 54
- 4S₇ = 54(1 − (1/3)⁷)/(1 − 1/3) = 81(1 − 1/2187) = 2186/27
- 5S∞ − S₇ = 81 − 2186/27 = 1/27
AnswerS∞ − S₇ = 81 − 2186/27 = 1/27 shown
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