How to use the summation symbol

If we want to add the expression $x_1,x_2$ all the way up to $x_{10}$ , it is quite cumbersome to write $x_1 +x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 +x_9 +x_{10}$ . Mathematical notations permit us to shorten such addition using the $\cdots$ symbol to denote “all the way up to” or “all the way down to”. Using the this symbol, the expression above can be written as $x_1 + x_2 + \cdots + x_{10}$ .

There is, however, a more compact way of writing sums. We can use the Greek letter $\Sigma$ as shown below.

Figure 1 – The Sigma Notation

In the figure above, a is the first index, and letter b is the last index. The variable(s) are the letters or the numbers that appear constantly in all terms. In the expression

$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10}$

$1$ is the first index, $10$ is the last index and $x$ is the variable. We use the letter $i$ as our index variable, or the variable that will hold the changing quantities. Hence, if we are going to use the sigma or the summation notation for the expression $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10}$ , we have

$\displaystyle\sum_{i=1}^{10} x_i$

Some of the examples are shown below. Observe the colors of the indices and the variables, to familiarize yourself how the summation symbol works.

Figure 2 – Examples of Summation Notation

In using the summation symbol, take note of the following:

An index variable is just a “dummy” variable. It means that you can use a different index variable without changing the value of the sum. The sum $\displaystyle\sum_{i=1}^{10} a_i$ is the same as $\displaystyle\sum_{j=1}^{10} a_j$ and is the same as $\displaystyle\sum_{k=1}^{10} a_k$ .
The indices are the natural numbers $1, 2, 3, \cdots$ and so on.
The last index is always greater than the first index.
A variable without an index most of the time represent an infinite sum or a sum from $1$ through $n$

More Examples

1.	$(a - 1) + (a^2 - 2) + (a^3 - 3) + (a^4 -4)$	$\displaystyle\sum_{i=1}^{4} (a^i - i)$
2.	$3p_5 + 3p_6 + 3p_7 + 3p_8$	$\displaystyle\sum_{j=5}^{8} 3p_j$
3.	$5 + 5 + 5 + 5 + 5 + 5 + 5$	$\displaystyle\sum_{k=1}^{7} 5$
4.	$1 + 2 + 3 + \cdots + 99 + 100$	$\displaystyle\sum_{m=1}^{100} m$
5.	$(a_3 + b_3) + (a_4 + b_4) +(a_5 + b_5)$	$\displaystyle\sum_{n=3}^{5} (a_n + b_n)$

Properties of the Summation Symbol

1.) The expression $(x_1 + x_2 + x_3 + x_4) + (x_5 + x_6 + x_7 + x_8 + x_9 + x_{10})$ equals $(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10})$ which means that $\displaystyle\sum_{i=1}^{4} x_i + \displaystyle\sum_{j=5}^{10} x_j = \displaystyle\sum_{i=1}^{10} x_i$ . In general, $\displaystyle\sum_{i=1}^{m} x_i + \displaystyle\sum_{j=m+1}^{n} x_j = \displaystyle\sum_{i=1}^{n} x_i$ .

2.)The expression $(x_1 + x_2 + x_3 + x_4) + (y_1 + y_2 + y_3 + y_4) = \displaystyle\sum_{i=1}^{4} x_i + \displaystyle\sum_{j=1}^{4} y_j$ . Regrouping the expression, we have $(x_1 + y_1) + (x_2 + y_2) + (x_3 + y_3) + (x_4 + y_4) = \displaystyle\sum_{i=1}^{4} (x_i + y_i)$ . This means that $\displaystyle\sum_{i=1}^{4} x_i + \displaystyle\sum_{i=j}^{4} y_j = \displaystyle\sum_{i=1}^{4} (x_i + y_i)$ Generalizing, we have $\displaystyle\sum_{i=1}^{n} x_i \pm \displaystyle\sum_{j=1}^{n} y_j = \displaystyle\sum_{i=1}^{n} (x_i \pm y_i).$

3.) The expression $c + c + c + \cdots + c$ ( $k$ of them) $= \displaystyle\sum_{i=1}^{k} c$ . But $c + c + c + \cdots + c = c( 1 + 1 + 1 + \cdots + 1)$ ( $k$ of them) $= kc$ . Therefore, $\displaystyle\sum_{i=1}^{k} c = kc$ .

4.) The expression $2x_1 + 2x_2 + 2x_3 + 2x_4 = \displaystyle\sum_{i=1}^{4} 2x_i$ . But $2x_1 + 2x_2 + 2x_3 + 2x_4 = 2(x_1 + x_2 + x_3 + x_4) = 2 \displaystyle\sum_{i=1}^{4} x_i$ . In general, $\displaystyle\sum_{i=1}^{k} cx_i = c \displaystyle\sum_{i=1}^{k} x_i$ .

16 comments on “How to use the summation symbol”

colonel davis on February 16, 2010 at 11:12 pm said:

?
∑ 3n-10= 175
n=1

Can you help me with this problem. Has something to do with partial sums.

Reply ↓

Guillermo Bautista on February 17, 2010 at 12:05 am said:

This is not a math tutorial site, but since you are the only one who asked, I’ll answer for now.
I don’t answer questions directly but I am going to give you a hint.

Try to see the pattern:

3(1) – 10 = -7
3(2) – 10 = – 4
3(3) – 10 = – 1

So basically, you have an arithmetic sequence with constant difference 3, first term -7 and sum of 175. I guess that hint is more than enough for you to solve the problem.

If you want math help, you should see this post. It has helped me in the past.

http://mathandmultimedia.com/2010/01/08/the-best-free-math-tutorial-website/

Reply ↓

misugrrl on April 18, 2010 at 9:34 pm said:

Just wanted to thank you for this post. You explained it in plain language for those of us who are mathematically challenged

Reply ↓

Guillermo Bautista on April 19, 2010 at 8:16 am said:

@misugrrl

You’re welcome. I am glad it has helped you.

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jennifer on May 6, 2010 at 8:20 pm said:

hi i was just wondering, how would you type that into an excel spreadsheet?

Reply ↓

Guillermo Bautista on May 7, 2010 at 8:15 am said:

I do not know of any way that you can write summation in excel. Maybe you should try writing it in MS Word first, the copy and paste to Excel.

Reply ↓
- sonia on February 3, 2011 at 9:26 pm said:
  
  hi! do u know if there is any way to calculate it in excel? imagine u have from n=1 to n=100… u cannot write it manually, thanks!
- Guillermo Bautista on February 4, 2011 at 8:27 am said:
  
  Well, as far as I know, excel has no summation function. The best thing that you can do is to place the numbers in excel (you don’t have to type them all, just use a formula), and then use the sum function.