Algebra 2
Sequences and series
Sigma notation
20 practice questions
0 video lessons
Theory + worked examples
Sigma Notation
California Algebra 2 • Standard A-SSE.1b • Sequences & Series
Sigma Notation is a topic in Sequences & Series in the California Common Core State Standards. It is aligned to Standard A-SSE.1b, which requires students to interpret expressions written in summation (sigma) notation and evaluate them.
Sigma notation \(\displaystyle\sum\) writes a sum compactly, with a start index below, an end index above, and a rule for each term.
Theory
Sigma notation \(\displaystyle\sum\) writes a sum compactly:
\[\sum_{k=1}^{n} a_k=a_1+a_2+\dots+a_n.\]
- The bottom is the starting index.
- The top is the ending index.
- The expression gives each term.
Substitute each index value and add the results.
\(\sum_{k=1}^{4}2k=2+4+6+8=20\).
The parts of sigma notation.
The meaning:
\[\sum_{k=1}^{n} a_k=a_1+a_2+\cdots+a_n\]
The number of terms is top minus bottom plus one.
How to evaluate a sum
- Read the start and end index.
- Substitute each index into the expression.
- List all the terms.
- Add them up.
Example 1 — Expand and evaluate
Evaluate \(\displaystyle\sum_{k=1}^{4} 2k\).
Solution
List the terms and add.
| \(2+4+6+8\) | ||
| \(=\) | \(20\) |
Example 2 — Squares
Evaluate \(\displaystyle\sum_{k=1}^{3} k^2\).
Solution
Square each index and add.
| \(1+4+9\) | ||
| \(=\) | \(14\) |
Example 3 — Write in sigma notation
Write \(3+5+7+9\) using sigma notation.
Solution
The terms are \(2k+1\) for \(k=1\) to \(4\).
| \(3+5+7+9\) | \(=\) | \(\sum_{k=1}^{4}(2k+1)\) |
Example 4 — Sum of a constant
Evaluate \(\displaystyle\sum_{k=1}^{5} 3\).
Solution
A constant is added \(5\) times.
| \(3\times5\) | ||
| \(=\) | \(15\) |
Common pitfalls
Include both endpoints: \(k=1\) to \(4\) is four terms.
Substitute the index everywhere it appears.
A constant term is added once per index value.
Frequently asked questions
What does sigma notation mean?
It is a compact way to write a sum of terms.
What are the numbers above and below sigma?
The ending index (top) and starting index (bottom).
How many terms are in the sum?
Top minus bottom plus one.
How do you evaluate \(\sum_{k=1}^{3}k\)?
\(1+2+3=6\).
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