Sigma Notation
In short: Sigma notation is a compact way to write a long sum: the Greek letter Σ, an index with its starting and ending values, and a rule that tells what to add at each step.
Algebra 2AlgebraA-SSE.B.4
What is Sigma Notation?
Sigma notation squeezes a long addition into one short symbol. Instead of writing out twenty terms, you name the rule once and say where the counting starts and stops.
In \(\sum_{k=m}^{n} a_k\), the letter \(k\) is the index, \(m\) is the lower limit, \(n\) is the upper limit, and \(a_k\) is the rule. Substitute \(m, m+1, \ldots, n\) in turn and add the results. The index is a placeholder, so \(k\) could just as well be \(i\) or \(j\) without changing the value.
How to spot it
- A large Greek capital letter sits in front of an expression, with small numbers above and below it.
- A problem describes a sum of many similar terms, such as the first 50 multiples of 4.
- A series formula is written compactly, with the index running from 1 or 0 up to n.
- You are asked to expand a summation into a list of terms or to write a list of terms as a summation.
The method
- Read the lower limit, the upper limit and the rule that follows the sigma.
- Substitute the lower limit for the index, then each whole number up to the upper limit, to get the terms.
- Calculate each term separately, watching the signs and the order of operations.
- Add the terms; to rewrite a list as a sum, find the pattern in the terms and name its rule.
Sigma Notation: solved example
Problem
Expand and evaluate \(\sum_{k=2}^{6} (3k^2 – k)\).
Solution
Substitute the values 2, 3, 4, 5, 6 into \(3k^2 – k\): \(10\), \(24\), \(44\), \(70\) and \(102\).
Add them: \(10 + 24 + 44 + 70 + 102 = 250\). The sum equals 250.
Try it
Question 1
Write \(5 + 10 + 15 + 20 + 25 + 30\) in sigma form and give its value.
Show the answer
\(\sum_{k=1}^{6} 5k = 5 \cdot 21 = 105\).
Question 2
Evaluate \(\sum_{i=1}^{4} 2^i\).
Show the answer
\(2 + 4 + 8 + 16 = 30\).
Sigma Notation vs. Arithmetic Series
Sigma notation is a way of writing a sum, and it can hold any rule at all: squares, powers, alternating signs. An arithmetic series is one specific kind of sum, with a constant step between terms. A series like that can be written with sigma, but not every sigma sum is arithmetic.
Keep going
Lessons
Worksheets and quizzes
- Arithmetic and Geometric Sequences and Series – Worksheet
- Arithmetic and Geometric Sequences and Series – Quiz
- Sequences and Series – Worksheet
- Sequences and Series – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Algebra