## 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, 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, $\displaystyle\sum_{i=1}^{k} cx_i = c \displaystyle\sum_{i=1}^{k} x_i$.