Sigma notation

Reading it

Σ_r=1^{5} (2r + 1) = 3 + 5 + 7 + 9 + 11 = 35.

The number of terms is (top - bottom + 1): Σ_r=4^{10} has 7 terms.

Spotting the series

• f(r) linear in r, such as 3r - 1: an arithmetic series
• f(r) like 5 x 2^r: a geometric series

Find a (the first term), d or r, and n, then use the sum formula.

Splitting and infinity

Σ_r=11^{20} f(r) = Σ_r=1^{20} f(r) - Σ_r=1^{10} f(r). An upper limit of ∞ means a sum to infinity, which exists for a geometric series with -1 < r < 1.

Worked example

Evaluate Σ_r=1^{20} (2r + 1).

1. First term (r = 1): 3. Last term (r = 20): 41.
2. 20 terms, arithmetic
3. S = (20/2)(3 + 41) = 10 x 44

Answer: 440

Check yourself

1. Evaluate Σ_r=1^{5} r².
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   55. 1 + 4 + 9 + 16 + 25 = 55.

2. Evaluate Σ_r=1^{12} (3r - 1).
   Show answer

   222. First term 2, last term 35, 12 terms: 6 x (2 + 35) = 222.

3. How many terms are in Σ_r=5^{30} f(r)?
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   26. 30 - 5 + 1 = 26.

4. Evaluate Σ_r=11^{20} r.
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   155. Σ_r=1^{20} r - Σ_r=1^{10} r = 210 - 55 = 155.

5. Evaluate Σ_r=1^{8} 2^r.
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   510. Geometric with first term 2, common ratio 2 and 8 terms: 2(2⁸ - 1)/(2 - 1) = 510. (See the geometric series lesson.)

6. Evaluate Σ_r=1^{∞} 6(1/3)^r.
   Show answer

   3. Geometric with first term 6 x 1/3 = 2 and common ratio 1/3. S_inf = 2/(1 - 1/3) = 3. (See the geometric series lesson.)

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 2 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.