All subjects · Further Mathematics · WFM01 Further Pure Mathematics 1

Sigma notation and standard sums

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

Sigma notation writes a sum compactly. With three standard results for the sums of r, r2 and r3, you can add up long series without adding term by term.

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Sigma notation

Σ_(r=1)n f(r) means f(1) + f(2) + ... + f(n). The letter r is a dummy variable.

Rules: Σ(f(r) + g(r)) = Σf(r) + Σg(r) and Σ k f(r) = k Σ f(r). But Σ f(r)g(r) is not Σf(r) × Σg(r).

A constant summed n times: Σ_(r=1)n 1 = n, so Σ_(r=1)n 4 = 4n, not 4.

Standard results

Learn Σ1 = n and Σr; the others are standard results you use directly.

Sums not starting at 1

Σ_(r=k)n f(r) = Σ_(r=1)n f(r) - Σ_(r=1)k-1 f(r). Subtract the sum up to k - 1, not up to k, or you lose the term r = k.

Worked example

Find Σ_(r=11)30 r2.

  1. Σ_(r=1)30 r2 = 30 × 31 × 61/6 = 9455.
  2. Σ_(r=1)10 r2 = 10 × 11 × 21/6 = 385.
  3. Subtract.

Answer: 9070

Key idea: Σr = n(n + 1)/2, Σr2 = n(n + 1)(2n + 1)/6, Σr3 = n2(n + 1)2/4. Split sums; Σ from k to n = Σ to n minus Σ to k - 1.

Check yourself

  1. Find Σ_(r=1)50 r.
    Show answer

    1275. 50 × 51/2 = 1275.

  2. Find Σ_(r=1)20 r2.
    Show answer

    2870. 20 × 21 × 41/6 = 2870.

  3. Find Σ_(r=1)10 r3.
    Show answer

    3025. 102 × 112/4 = 3025 (= 552).

  4. Find Σ_(r=11)30 r2.
    Show answer

    9070. Σ_130 r2 - Σ_110 r2 = 9455 - 385 = 9070.

  5. Find Σ_(r=1)20 (3r - 2).
    Show answer

    590. 3Σr - 2Σ1 = 3(210) - 2(20) = 630 - 40 = 590.

  6. Which expression equals Σ_(r=k)n r?
    • n(n + 1)/2 - (k - 1)k/2
    • n(n + 1)/2 - k(k + 1)/2
    • (n - k)(n - k + 1)/2
    • n(n + 1)/2 - k
    Show answer

    n(n + 1)/2 - (k - 1)k/2. Subtract the sum from 1 to k - 1, which is (k - 1)k/2.

  7. A student writes Σ_(r=1)n 5 = 5. What is the correct value?
    • 5
    • 5n(n + 1)/2
    • 5n
    • n + 5
    Show answer

    5n. You add 5 once for each of the n values of r.

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Last updated 8 November 2026. Spotted a mistake? Tell us.