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4MA1/1H Jun 2021, Question 23: Sum of an Arithmetic Series

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1H (Higher) · Calculator · 5 marks

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The question, in short

An arithmetic series has first N terms summing to 292, with 2nd term 8.5 and 5th term 13. Find N, showing algebraic working.

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

Solution · 5 marks

  1. 1a + d = 8.5
  2. 2a + 4d = 13
  3. 3Subtracting: (a + 4d) − (a + d) = 13 − 8.5
  4. 43d = 4.5
  5. 5d = 1.5
  6. 6Substitute d = 1.5 into a + d = 8.5
  7. 7a + 1.5 = 8.5
  8. 8a = 7
  9. 9Sum of first N terms: S_N = N/2 [2a + (N − 1)d]
  10. 10S_N = N/2 [2(7) + (N − 1)(1.5)] = 292
  11. 11N/2 [14 + 1.5(N − 1)] = 292
  12. 12N/2 [14 + 1.5N − 1.5] = 292
  13. 13N/2 [12.5 + 1.5N] = 292
  14. 14N(12.5 + 1.5N)/2 = 292
  15. 15N(1.5N + 12.5) = 584
  16. 161.5N² + 12.5N − 584 = 0
  17. 17Multiply by 2: 3N² + 25N − 1168 = 0
  18. 18Use quadratic formula: N = [-25 ± √(25² − 4×3×(−1168))]/(2×3)
  19. 19N = [-25 ± √(625 + 14016)]/6
  20. 20N = [-25 ± √14641]/6
  21. 21√14641 = 121
  22. 22N = (−25 + 121)/6 or N = (−25 − 121)/6
  23. 23N = 96/6 = 16 or N = −146/6 (discard negative)
  24. 24N = 16

Answer16

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