SUMMATION NOTATION
LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
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Summation notation gives a compact way to represent sums, when the terms exhibit some common pattern.

For example, consider this sum: $$\cssId{s3}{1+4+9+16+25+36+49+64+81+100}$$ Each term is a perfect square.
Re-write the sum to clearly show the pattern: $$ \cssId{s6}{1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2} $$ Using summation notation, the five most common representations for this sum are: $$ \cssId{s8}{\sum_{i=1}^{10}\ i^2} \ \ \ \ \cssId{s9}{\text{ or } \ \ \ \ \sum_{j=1}^{10}\ j^2} \ \ \ \ \cssId{s10}{\text{ or } \ \ \ \ \sum_{k=1}^{10}\ k^2} \ \ \ \ \cssId{s11}{\text{ or } \ \ \ \ \sum_{m=1}^{10}\ m^2} \ \ \ \ \cssId{s12}{\text{ or } \ \ \ \ \sum_{n=1}^{10}\ n^2} $$ Summation notation is also called sigma notation.

COMMENTS ON SUMMATION NOTATION

Using $\sum_{i=1}^{10}\ i^2\,$ as an example:

EXAMPLES:
$\displaystyle \cssId{s36}{\sum_{j = -1}^2\ j^3} \cssId{s37}{= \overset{j=-1}{\overbrace{(-1)^3}} + \overset{j=0}{\overbrace{\ 0^3\ }} +\overset{j=1}{\overbrace{\ 1^3\ }} + \overset{j=2}{\overbrace{\ 2^3\ }}} \cssId{s38}{= -1 + 0 + 1 + 8} \cssId{s39}{= 8} $
$\displaystyle \cssId{s40}{\sum_{k = 3}^7\ x_k} \cssId{s41}{= x_3 + x_4 + x_5 + x_6 + x_7} $
$\displaystyle \cssId{s42}{\sum_{n = -2}^3\ 5} \cssId{s43}{= \overset{n=-2}{\overbrace{\ \ 5\ \ }} + \overset{n=-1}{\overbrace{\ \ 5\ \ }} + \overset{n=0}{\overbrace{\ \ 5\ \ }} + \overset{n=1}{\overbrace{\ \ 5\ \ }} + \overset{n=2}{\overbrace{\ \ 5\ \ }} + \overset{n=3}{\overbrace{\ \ 5\ \ }}} \cssId{s44}{= 6\cdot 5} \cssId{s45}{= 30} $
PROPERTIES OF SUMS
Let $\,a\,$ and $\,b\,$ be integers with $\,a\lt b\,$, and let $\,k\,$ be any real number.

You can split sums and differences apart: $$ \begin{gather} \cssId{s51}{\sum_{i=a}^b\ (x_i + y_i)} \cssId{s52}{= \sum_{i=a}^b\ x_i + \sum_{i=a}^b\ y_i}\cr\cr \cssId{s53}{\sum_{i=a}^b\ (x_i - y_i)} \cssId{s54}{= \sum_{i=a}^b\ x_i - \sum_{i=a}^b\ y_i} \end{gather} $$
You can slide a constant out of a sum:
$$ \cssId{s56}{\sum_{i=a}^b\ kx_i} \cssId{s57}{= k\ \sum_{i=a}^b\ x_i } $$ To sum a constant value, you must correctly count the number of terms: $$ \cssId{s59}{\sum_{i=a}^b\ k} \cssId{s60}{= (b-a+1)\cdot k} $$
WHY DO THESE PROPERTIES WORK?

The following examples illustrate why these properties work.
They are all easy consequences of the commutative (re-ordering) and associative (re-grouping) properties of addition,
and the distributive law:

Master the ideas from this section
by practicing the exercise at the bottom of this page.


When you're done practicing, move on to:
Mean, Median, and Mode


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11
AVAILABLE MASTERED IN PROGRESS

(MAX is 11; there are 11 different problem types.)