Sum

The sum of two numbers is their value added together. This operation is called additive summation or addition. There are many ways of writing sums, including:

• Addition (${\displaystyle 2+4+6=12}$)
• Summation (${\displaystyle \sum _{k=1}^{3}k=1+2+3=6}$)
• Code:

Sum = 0

For I = M to N

Sum = Sum + X(I)

Next I (in Visual BASIC)

Sigma notation

Sigma notation is a mathematical notation to write long sums in a short way. Sigma notation uses the Greek letter Sigma (${\displaystyle \Sigma }$), and takes upper and lower bounds which tell us where the sum begins and where it ends. The lower bound usually has a variable (called the index, often denoted by ${\displaystyle i}$, ${\displaystyle j}$ or ${\displaystyle k}$^[1]) along with a value, such as "${\displaystyle i=2}$". This tells us that the summation begins at 2, and goes up by 1 until it reaches the number on the top.^[2]

Properties

${\displaystyle \sum _{i=1}^{n}0=0}$

${\displaystyle \sum _{i=1}^{n}1=n}$

${\displaystyle \sum _{i=1}^{n}n=n^{2}}$

${\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}}$^[3]

${\displaystyle \sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}}$^[3]

${\displaystyle \sum _{i=1}^{n}i^{3}={\frac {n^{2}(n+1)^{2}}{4}}}$^[3]

${\displaystyle \sum _{i=1}^{\infty }a_{i}=\lim _{t\to \infty }\sum _{i=1}^{t}a_{i}}$

Applications

Sums are used to represent series and sequences. For example:

${\displaystyle \sum _{i=1}^{4}{\frac {1}{2^{i}}}={\frac {1}{2^{1}}}+{\frac {1}{2^{2}}}+{\frac {1}{2^{3}}}+{\frac {1}{2^{4}}}}$

The geometric series of a repeating decimal can be represented in summation. For example:

${\displaystyle \sum _{i=1}^{\infty }{\frac {3}{10^{i}}}=0.333333...={\frac {1}{3}}}$

The concept of an integral is a limit of sums, with the area under a curve being defined as:

${\displaystyle \lim _{n\to \infty }\sum _{i=1}^{n}f(x_{i}^{*})\Delta x}$

Calculating the mean of a set of numbers can be expressed like this, where each ${\displaystyle x}$ can be any number.

${\displaystyle {\frac {1}{|m-n|}}\sum _{i=m}^{n}x_{i}}$

So, because ${\displaystyle \sum _{i=m}^{n}{\frac {i}{2}}={\frac {1}{2}}\sum _{i=m}^{n}i}$, you could also calculate the mean like ${\displaystyle \sum _{i=m}^{n}{\frac {x_{i}}{|m-n|}}}$ as well.

Related pages

• Product (mathematics)