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4PM1/01 Jun 2024, Question 3: Sigma Notation

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 8 marks

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The question, in short

Prove the summation formula for 5r − 3 from r = 1 to n, then use it to evaluate a sum from r = 31 to 60 and to find n when the total equals 3783.

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

Part (a) · 3 marks

Show that the sum of 5r − 3 for r from 1 to n equals (n/2)(5n − 1).

  1. 1The series is arithmetic with a = 2, d = 5
  2. 2Sₙ = (n/2)[2 × 2 + (n − 1) × 5]
  3. 3= (n/2)(4 + 5n − 5)
  4. 4= (n/2)(5n − 1)

Answer∑ from r = 1 to n of (5r − 3) = (n/2)(5n − 1) shown

Part (b) · 2 marks

Using part a or otherwise, evaluate the sum of 5r − 3 for r from 31 to 60.

  1. 1Sum from 31 to 60 = S₆₀ − S₃₀
  2. 2S₆₀ = (60/2)(5 × 60 − 1) = 8970
  3. 3S₃₀ = (30/2)(5 × 30 − 1) = 2235
  4. 48970 − 2235 = 6735

Answer6735

Part (c) · 3 marks

Given the sum of 5r − 3 from r = 1 to n is 3783, find the value of n.

  1. 1(n/2)(5n − 1) = 3783
  2. 25n² − n − 7566 = 0
  3. 3(5n + 194)(n − 39) = 0
  4. 4n = 39 (n = −194/5 is not valid)

Answern = 39

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