SUMMATION NOTATION
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Summation notation gives a compact way to represent sums, when the terms exhibit some common pattern.

For example, consider the sum [beautiful math coming... please be patient] $$1+4+9+16+25+36+49+64+81+100\,,\,$$ which can be rewritten as: [beautiful math coming... please be patient] $$ 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 $$ This sum can be represented using summation notation as: [beautiful math coming... please be patient] $$ \sum_{i=1}^{10}\ i^2 \ \ \ \ \text{ or } \ \ \ \ \sum_{j=1}^{10}\ j^2 \ \ \ \ \text{ or } \ \ \ \ \sum_{k=1}^{10}\ k^2 \ \ \ \ \text{ or } \ \ \ \ \sum_{m=1}^{10}\ m^2 \ \ \ \ \text{ or } \ \ \ \ \sum_{n=1}^{10}\ n^2 $$ Summation notation is also called sigma notation.

COMMENTS ON SUMMATION NOTATION

Using [beautiful math coming... please be patient] $\sum_{i=1}^{10}\ i^2\,$ as an example:

EXAMPLES:
[beautiful math coming... please be patient] $\displaystyle \sum_{j = -1}^2\ j^3 = \overset{j=-1}{\overbrace{(-1)^3}} + \overset{j=0}{\overbrace{\ 0^3\ }} +\overset{j=1}{\overbrace{\ 1^3\ }} + \overset{j=2}{\overbrace{\ 2^3\ }} = -1 + 0 + 1 + 8 = 8 $
[beautiful math coming... please be patient] $\displaystyle \sum_{k = 3}^7\ x_k = x_3 + x_4 + x_5 + x_6 + x_7 $
[beautiful math coming... please be patient] $\displaystyle \sum_{n = -2}^3\ 5 = \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\ \ }} = 6\cdot 5 = 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: [beautiful math coming... please be patient] $$ \begin{gather} \sum_{i=a}^b\ (x_i + y_i) = \sum_{i=a}^b\ x_i + \sum_{i=a}^b\ y_i\cr\cr \sum_{i=a}^b\ (x_i - y_i) = \sum_{i=a}^b\ x_i - \sum_{i=a}^b\ y_i \end{gather} $$
You can slide a constant out of a sum:
[beautiful math coming... please be patient] $$ \sum_{i=a}^b\ kx_i = k\ \sum_{i=a}^b\ x_i $$ To sum a constant value, you must correctly count the number of terms: [beautiful math coming... please be patient] $$ \sum_{i=a}^b\ k = (b-a+1)\cdot k $$
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.
(MAX is 11; there are 11 different problem types.)