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MathidiaMathematics Academy
Further series
Lesson 21 of 75
Introduction

Standard finite sums

Objective. Use formulae for sums of powers.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Standard finite sums provide ready-made formulae for sums of powers of the first n positive integers: Σr = n(n+1)/2, Σr² = n(n+1)(2n+1)/6, and Σr³ = [n(n+1)/2]². These formulae allow sums of polynomial expressions in r to be evaluated exactly by splitting the sum into separate standard sums and combining with constants factored out. For example, Σ(2r²+3r) from r=1 to n equals 2Σr² + 3Σr, which can be substituted and simplified into a single polynomial in n.

Worked example

Find Σ(3r² − r) for r = 1 to n, giving your answer as a fully factorised expression in n.

Find Σ(3r² − r) for r = 1…
Illustrated card showing Find Σ(3r² − r) for r = 1 to n
  1. 1Split the sum: Σ(3r²−r) = 3Σr² − Σr
  2. 2Substitute standard sums: 3×n(n+1)(2n+1)/6 − n(n+1)/2
  3. 3Simplify first term: n(n+1)(2n+1)/2 − n(n+1)/2
  4. 4Factor out n(n+1)/2: n(n+1)/2 × [(2n+1) − 1] = n(n+1)/2 × 2n = n²(n+1)

Answer: n²(n+1)

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Common mistake. Students apply the formula Σr² to expressions like Σ(r+1)² without first expanding the bracket, treating it as if r² and (r+1)² sum the same way. Always expand brackets fully before splitting into standard sums.

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