Sigma Notation (Σ) Explained: How to Read and Write Summations

Q: What is the sum of 1 to n?

Σ_k=1^n k = n(n + 1)/2. For example 1 + 2 + … + 100 = 5050.

How to read the parts of a sigma expression, expand and evaluate one, write a sum in sigma form, and use the standard formulas for adding up $1, 2, 3, \dots$

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Writing $1 + 4 + 9 + 16 + \dots + 100$ is tiring and a little vague. Sigma notation describes the same sum exactly, in a few symbols: $\sum_{k=1}^{10} k^2$ .

Reading it

\sum_{k=1}^{4} (2k + 1)

Read: "the sum, from $k = 1$ to $4$ , of $2k + 1$ ".

$\Sigma$ (capital sigma, the Greek S) means sum. The letter underneath, $k$ , is the index: it counts through whole numbers. The bottom gives its starting value and the top its last value. The expression to the right says what to add for each value of the index.

$\sum_{k=1}^{4} (2k + 1)$
Let $k$ run from 1 to 4.

1/4

Writing a sum in sigma form

Look for the pattern that turns the term number into the term, then choose where the index starts and stops.

Worked example

Write $2 + 4 + 6 + \dots + 40$ in sigma notation.

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Each term is $2k$ for $k = 1, 2, 3, \dots$

Tip

The index letter does not matter. $\sum_{k=1}^{20} 2k$ and $\sum_{i=1}^{20} 2i$ are the same sum.

Useful rules and formulas

Useful sums

\sum_{k=1}^{n} k = \frac{n(n + 1)}{2} \qquad \sum_{k=1}^{n} c = cn

Sums also split and scale like ordinary addition: $\sum (a_k + b_k) = \sum a_k + \sum b_k$ , and $\sum c\,a_k = c \sum a_k$ .

Worked example

Evaluate $\sum_{k=1}^{50} (3k - 2)$ .

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Split and scale.
$3\sum_{k=1}^{50} k - \sum_{k=1}^{50} 2$

Watch out

Count the terms carefully. $\sum_{k=3}^{10}$ has $10 - 3 + 1 = 8$ terms, not $7$ .

Key idea

Sigma notation is everywhere in calculus: integrals are defined as limits of sums of thin rectangles, and those sums are written with $\Sigma$ .

Common questions

What does the number under sigma mean?: It is the starting value of the index. The number on top is the last value. The index steps up by one each time.
What is the sum of 1 to n?: $\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}$ . For example $1 + 2 + \dots + 100 = 5050$ .

Practise sigma notation

The Sigma Notation lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What does the number under sigma mean?

It is the starting value of the index. The number on top is the last value. The index steps up by one each time.

What is the sum of 1 to n?

$\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}$ . For example $1 + 2 + \dots + 100 = 5050$ .