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Pre-Calculus Sequences and series (advanced)

Sigma notation (advanced)

20 practice questions 0 video lessons Theory + worked examples

Sigma Notation

Common Core Pre-Calculus • Standard A-SSE.4 • Sequences & Series

Sigma Notation is a topic in Sequences & Series in the Common Core State Standards. It is aligned to Standard A-SSE.4, which requires students to use sigma notation to represent and evaluate sums.

Sigma notation \(\displaystyle\sum_{k=1}^{n}a_k\) writes a sum compactly, with the index running from a lower to an upper limit.

Common Core Pre-Calculus › Sequences & Series › Sigma Notation  —  Standard A-SSE.4

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Theory

Sigma notation writes a sum compactly using the Greek letter \(\Sigma\):
\[\sum_{k=1}^{n}a_k=a_1+a_2+\cdots+a_n.\]

The index \(k\) starts at the lower limit and runs to the upper limit, and \(a_k\) is the rule for each term.

Sums are linear: you can factor out constants and split sums apart.

Handy formulas: \(\displaystyle\sum_{k=1}^{n}k=\dfrac{n(n+1)}{2}\) and \(\displaystyle\sum_{k=1}^{n}c=cn\).
Sigma notation Sigma notation compactly writes a sum: the index starts at the bottom value and runs to the top value. ∑nk=1aₖ= a₁+a₂+…+aₙupper limitlower limit & index
Reading sigma notation: index, limits, and term rule.
Sum properties Sum properties Sum properties ∑ c·aₖ = c ∑ aₖ ∑(aₖ+bₖ) = ∑aₖ + ∑bₖ ∑ₙ c = c·n
The linearity and constant properties.

Properties and two standard sums:

\[\sum c\,a_k=c\sum a_k,\qquad \sum(a_k+b_k)=\sum a_k+\sum b_k\]
\[\sum_{k=1}^{n}k=\dfrac{n(n+1)}{2},\qquad \sum_{k=1}^{n}c=cn\]
constants factor out of a sum; sums split over addition; the sum of 1 to n is n times n plus 1 over 2
Change the index carefully if you shift the limits — the term rule must move with it.

How to evaluate a sum

  1. Read the index, its limits, and the term rule.
  2. Expand for a few terms, or apply a formula for many.
  3. Use properties to factor constants and split sums.
  4. Add or apply \(\dfrac{n(n+1)}{2}\), \(cn\), etc.
Example 1 — Expand a sum
Write out \(\displaystyle\sum_{k=1}^{4}(2k+1)\) and evaluate.
Solution

Substitute \(k=1,2,3,4\) and add.

\(=\)3+5+7+9
\(=\)24
the sum is 24
Example 2 — A constant sum
Evaluate \(\displaystyle\sum_{k=1}^{10}7\).
Solution

Adding a constant \(n\) times gives \(cn\).

\(=\)7\cdot 10=70
the sum is 70
Example 3 — Use a known formula
Evaluate \(\displaystyle\sum_{k=1}^{100}k\).
Solution

Use \(\displaystyle\sum_{k=1}^{n}k=\dfrac{n(n+1)}{2}\).

\(=\)\dfrac{100\cdot 101}{2}
\(=\)5050
the sum of 1 to 100 is 5050
Example 4 — Split with properties
Evaluate \(\displaystyle\sum_{k=1}^{5}(3k)\).
Solution

Factor out the constant, then sum.

\(=\)3\sum_{k=1}^{5}k
\(=\)3\cdot 15=45
the sum is 45

Common pitfalls

Include both limits. \(\sum_{k=1}^{4}\) has four terms, \(k=1,2,3,4\).
Constants still get summed. \(\sum_{k=1}^{n}c=cn\), not \(c\).
Factor constants, don't drop them. \(\sum 3k=3\sum k\).

Frequently asked questions

What is sigma notation?

A compact way to write a sum, \(\sum_{k=1}^{n}a_k\), where the index \(k\) runs from the lower to the upper limit.

How do you expand a sigma sum?

Substitute each index value from the lower to the upper limit and add the results.

Can you factor a constant out of a sum?

Yes: \(\sum c\,a_k=c\sum a_k\). Sums are linear.

What is the sum of the first n whole numbers?

\(\sum_{k=1}^{n}k=\dfrac{n(n+1)}{2}\).

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