Key takeaway: Sigma notation uses the Greek capital letter Σ with an index, a start value and an end value to write a long sum compactly, such as Σ(2k + 1) for k = 3 to 7.
Grade 11 – Algebra 2A-SSE.B.4Expressions and Equations
Definition of Sigma Notation
Sigma notation is shorthand for adding many terms that follow one rule. The capital Greek letter sigma means “add up”. Beneath it sits the index variable and its starting value; above it sits the final value; to its right sits the rule that produces each term.
To read it, plug each whole number from the start to the end into the rule, one at a time, and add the results. The index is only a placeholder, so renaming it from \(k\) to \(i\) changes nothing. A sum with a hundred terms fits in one short line of sigma notation.
Formula and notation
The general pattern, with index \(k\), lower limit \(m\) and upper limit \(n\):
\[\sum_{k=m}^{n} f(k)=f(m)+f(m+1)+\cdots+f(n).\]
The number of terms is \(n-m+1\). Constants can be pulled out front, and a sum of two rules splits into two separate sums.
Key facts
- The count of terms is upper minus lower plus 1, not upper minus lower.
- The index may start at 0, 1, or any whole number the problem chooses.
- A constant factor can move outside the sigma without changing the total.
- Arithmetic and geometric series both have a compact form in sigma notation.
Sigma Notation example
Evaluate \(\sum_{k=3}^{7}(2k+1)\).
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List the index values: \(k=3,4,5,6,7\). That is \(7-3+1=5\) terms.
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Substitute each into \(2k+1\): 7, 9, 11, 13, 15.
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Add: \(7+9+11+13+15=55\).
The sum equals 55.
Common mistake
Watch out
Leona claims \(\sum_{k=3}^{7}(2k+1)\) has \(7-3=4\) terms and adds only 7, 9, 11 and 13 to get 40.
Why it is wrong
Both endpoints are included, so the count is \(7-3+1=5\). The missing fifth term is 15, and 40 + 15 = 55. In sigma notation, count with both endpoints.
Related terms and lessons
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