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Summing series with the standard results

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

To sum a series such as Σ r(r + 2), expand the general term into powers of r, apply the standard results to each part, and then factorise the answer.

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Method

1. Expand the general term: r(r + 2) = r2 + 2r.

2. Split: Σ r2 + 2Σ r.

3. Substitute the standard results.

4. Take out common factors early, usually n/6 or n(n + 1)/6, and simplify what is left.

Example of the algebra

Σ_(r=1)n r(r + 2) = n(n + 1)(2n + 1)/6 + 2 × n(n + 1)/2

= n(n + 1)/6 × [(2n + 1) + 6]

= n(n + 1)(2n + 7)/6.

Check with n = 1: the sum is 1 × 3 = 3, and the formula gives 1 × 2 × 9/6 = 3.

Sums with n in the limits

For Σ_(r=n+1)2n, work out Σ_(r=1)2n by replacing n with 2n in the formula, then subtract Σ_(r=1)n.

Always check small cases (n = 1, n = 2) against your final expression.

Worked example

Show that Σ_(r=1)n (2r - 1)2 = n(2n - 1)(2n + 1)/3.

  1. (2r - 1)2 = 4r2 - 4r + 1.
  2. 4 × n(n + 1)(2n + 1)/6 - 4 × n(n + 1)/2 + n
  3. = n/3 × [2(n + 1)(2n + 1) - 6(n + 1) + 3] = n/3 × (4n2 - 1).

Answer: n(2n - 1)(2n + 1)/3

Key idea: Expand the term, split the sum, use the standard results, factorise early. Check with n = 1.

Check yourself

  1. Find Σ_(r=1)10 r(r + 2).
    Show answer

    495. Using n(n + 1)(2n + 7)/6 with n = 10: 10 × 11 × 27/6 = 495.

  2. Find Σ_(r=1)10 (2r - 1)2.
    Show answer

    1330. n(2n - 1)(2n + 1)/3 with n = 10: 10 × 19 × 21/3 = 1330.

  3. Which is Σ_(r=1)n r(r + 2) in factorised form?
    • n(n + 1)(2n + 1)/6
    • n(n + 1)(2n + 7)/6
    • n(n + 1)(n + 2)/3
    • n(n + 1)(2n + 5)/6
    Show answer

    n(n + 1)(2n + 7)/6. Σr2 + 2Σr = n(n + 1)/6 × [(2n + 1) + 6].

  4. Σ_(r=1)n r(r + 1)(r - 1) = n(n + 1)(n - 1)(n + 2)/4. Use this to find the sum when n = 6.
    Show answer

    420. 6 × 7 × 5 × 8/4 = 420.

  5. Find Σ_(r=11)20 r2 using Σ_(r=n+1)2n r2 = n(2n + 1)(7n + 1)/6.
    Show answer

    2485. With n = 10: 10 × 21 × 71/6 = 2485.

  6. A student writes Σ r(r + 2) = Σ r × Σ (r + 2). What is wrong?
    • Nothing is wrong
    • They should have added the sums instead
    • The sum of a product is not the product of the sums
    • They should have used Σ r3
    Show answer

    The sum of a product is not the product of the sums. Check with n = 2: Σ r(r + 2) = 3 + 8 = 11, but Σr × Σ(r + 2) = 3 × 7 = 21.

  7. Σ_(r=1)n (r2 + 4r) = n(n + 1)(2n + a)/6. Find a.
    Show answer

    13. n(n + 1)(2n + 1)/6 + 4n(n + 1)/2 = n(n + 1)/6 × [(2n + 1) + 12] = n(n + 1)(2n + 13)/6.

  8. Find Σ_(r=1)10 r2(r + 2), using Σ_(r=1)n r3 = n2(n + 1)2/4 and Σ_(r=1)n r2 = n(n + 1)(2n + 1)/6.
    Show answer

    3795. r2(r + 2) = r3 + 2r2, so the sum is Σr3 + 2Σr2 = 3025 + 2 × 385 = 3795.

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