Sigma Notation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sigma Notation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A compact way to write the sum of many terms using the Greek letter \Sigma (sigma). \sum_{i=m}^{n} a_i means add up a_i for every integer i from m to n.

Sigma notation is shorthand for 'add these all up.' The letter below \Sigma is a counter, the number below is where to start, the number above is where to stop, and the expression to the right tells you what to add each time.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Sigma notation is the language of summation. It turns long, repetitive sums into compact, precise expressions and is essential for stating formulas for series, statistics, and linear algebra.

Common stuck point: The index variable is a dummy—\sum_{i=1}^{n} i^2 and \sum_{k=1}^{n} k^2 are identical. The variable disappears after summation, just like a definite integral's variable.

Sense of Study hint: Expand the sum by writing out the first three terms and the last term to see the pattern before simplifying.

Worked Examples

Example 1

easy
Expand and evaluate \displaystyle\sum_{k=1}^{5} (2k - 1).

Solution

  1. 1
    Write out each term: k=1: 1, k=2: 3, k=3: 5, k=4: 7, k=5: 9.
  2. 2
    Sum: 1+3+5+7+9 = 25.
  3. 3
    Alternatively, use linearity: 2\sum_{k=1}^5 k - \sum_{k=1}^5 1 = 2 \cdot 15 - 5 = 25.

Answer

25
Expanding by substituting each value of k is the most direct approach. The linearity of \Sigma allows splitting the sum and using the formula \sum_{k=1}^n k = \frac{n(n+1)}{2}.

Example 2

medium
Write 1^2 + 2^2 + 3^2 + \cdots + n^2 in sigma notation and evaluate the closed form for n = 10.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Evaluate \displaystyle\sum_{i=0}^{4} 3^i.

Example 2

medium
Rewrite \displaystyle\sum_{j=1}^{n}(3j^2 + 2j - 1) using linearity of summation.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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