IB Mathematics: Analysis and Approaches SL/HL — Topic 1 Number and algebra
Reading and Writing Sigma Notation
A concise reference for unpacking summation notation, expanding finite sums and writing a pattern compactly using the sigma symbol.
Learning goal
Read the parts of a sigma expression, expand it term by term and write a finite pattern in compact sigma notation.
Syllabus link
Sigma notation is used in SL 1.3 when writing finite geometric sums.
Big idea
Sigma notation is simply a compact instruction for generating terms and adding them.
Number of terms
From index value \(a\) to \(b\), inclusive, there are \(b-a+1\) terms.
This page is a refresher. The geometric-sequences lesson uses sigma notation as part of finite geometric sums; use this reference whenever you need to unpack what a sigma expression means or build one from a pattern.
The anatomy of sigma notation
A sum written in sigma notation has the form
Read this as: “the sum, from \(k=a\) to \(k=b\), of \(f(k)\).”
What each part tells you
The lower line gives the index and its first value, the value above \(\Sigma\) gives the upper limit, and the expression to the right gives the general term to evaluate and add.
Expanding a sigma expression
Worked example 1: expand a geometric sigma sum
Expand
\[\sum_{k=1}^{5}4(2)^{k-1}.\]Here \(k=1,2,3,4,5\).
So the expression represents the first five terms of a geometric series with \(u_1=4\) and \(r=2\).
Writing a geometric series in sigma notation
Worked example 2: write the first \(n\) terms compactly
Write
\[6+18+54+\cdots+6(3)^{n-1}\]in sigma notation.
The kth term is \(6(3)^{k-1}\).
The first term corresponds to \(k=1\), and the nth term corresponds to \(k=n\).
Starting the index somewhere other than 1
Worked example 3: identify the first term and the number of terms
Consider
\[\sum_{k=3}^{8}5(1.2)^k.\]The first term occurs when \(k=3\):
\[u_1^{(\mathrm{sum})}=5(1.2)^3.\]The last term occurs when \(k=8\), and the number of terms is
\[\boxed{8-3+1=6}.\]Each increase of \(k\) multiplies the term by 1.2, so the common ratio is \(\boxed{1.2}\).
Number of terms
\[\boxed{\text{number of terms}=b-a+1}\]for a sum running from \(k=a\) to \(k=b\), inclusive.
The index letter is a dummy variable
The letter itself does not matter
The letters \(k\), \(r\), \(j\) or \(i\) can all be used as the summation index. Renaming the index does not change the value of the sum, provided the limits and general term are changed consistently.
For example,
\[\sum_{k=1}^{5}4(2)^{k-1}=\sum_{j=1}^{5}4(2)^{j-1}.\]The pattern and the limits are unchanged; only the name of the index has changed.
Using sigma notation with geometric sums
For a geometric sequence with first term \(u_1\) and common ratio \(r\), the sum of the first \(n\) terms can be written
This is the notation used in the geometric sequences and series lesson. The sigma expression generates the same terms as
\[u_1+u_1r+u_1r^2+\cdots+u_1r^{n-1}.\]Return to geometric sequences and series
Use this notation in context on the full SL 1.3 lesson, including finite sums, block sums and cumulative thresholds.
This reference page is separated from Appendix A of the finalized Radford Mathematics “SL 1.3 – Geometric Sequences and Series – AA SL-HL” handout. The annotated sigma diagram is a clean crop from the finalized PDF; ordinary mathematics and worked examples are recreated as accessible HTML + MathJax.