Sigma notation: what Σ means and how to work out a sum

Sigma notation is a short way to write a sum. The Greek capital letter Σ (sigma) means "add up". Below and above it you write where the counter starts and stops, and after it the rule for each term. For example, Σ from r = 1 to 5 of r² means 1² + 2² + 3² + 4² + 5² = 55.

How to read sigma notation

• r is the counter (sometimes i, k or n). It goes up in steps of 1.
• The bottom number is where r starts; the top number is where it stops.
• f(r) is the rule: substitute each value of r and add the results.
• Number of terms: from r = a to r = b there are b − a + 1 terms.

AQA's A-level Maths specification (content D3) asks you to understand and use sigma notation for sums of series.

Evaluating a sum

Example 1: write out the terms

Example 2: a sum that starts at zero

Remember 3⁰ = 1. Starting at r = 0 gives 4 terms here, not 3.

Rules for sigma notation

These rules let you split a sum into easier pieces:

• Constants come out: Σ k f(r) = k Σ f(r).
• Sums split: Σ (f(r) + g(r)) = Σ f(r) + Σ g(r).
• A constant on its own: Σ from r = 1 to n of c = cn, because you add c, n times.
• The sum of the first n whole numbers: Σ from r = 1 to n of r = ½n(n + 1).

For example, Σ from r = 1 to 10 of (2r + 1) = 2 × Σr + Σ1 = 2 × 55 + 10 = 120.

Changing the starting value

To add r from 5 to 12, take the sum up to 12 and subtract the sum up to 4: Σ from 1 to 12 of r − Σ from 1 to 4 of r = 78 − 10 = 68.

Arithmetic series in sigma notation

If the rule is linear in r, like 3r − 1, the terms go up by the same amount each time: an arithmetic series. Use the sum formula with the first term a, the last term l and the number of terms n:

Here the first term is 3(1) − 1 = 2, the last is 3(20) − 1 = 59 and there are 20 terms.

Geometric series in sigma notation

If r is a power, like 2ʳ, each term is multiplied by the same ratio: a geometric series. With first term a, ratio k and n terms:

Sigma notation in statistics

You will meet Σ in statistics too, where it means "add up all the values". The mean is x̄ = Σx ÷ n, and the standard deviation formulas use Σx² and Σ(x − x̄)². Our standard deviation calculator shows each sum.

Common mistakes

• Miscounting the terms. From r = 0 to 3 is 4 terms; use b − a + 1.
• Taking the constant out of the wrong place. Σ(2r + 1) is 2Σr + Σ1, not 2Σr + 1.
• Starting from 1 when the sum starts elsewhere. Subtract the terms you don't want.
• Using the arithmetic formula on a geometric sum. Check whether r is multiplied or is a power.

Practise with a short timed paper

Sequences, series and sigma notation are in A-level Maths. The daily A-level Maths: Algebra & Functions paper is a short timed paper of 8 questions, with a new one every day. Check sums in our sequences and series calculator.

Read next

Maths · AlgebraArithmetic series formula: Sₙ = n/2(2a + (n − 1)d), proofThe arithmetic series formula Sₙ = n/2[2a + (n − 1)d] and Sₙ = n/2(a + l): the proof, sigma notation, finding n, two-condition problems and examples.