# Sigma Notation: An Introduction With Its Formula, Properties, and Calculations

# Sigma Notation: An Introduction With Its Formula,
Properties, and Calculations

Summation notation is used to write the sum of
a finite sequence of a number in short form. A summation that represents the
sum of an infinite sequence is known as a series. Summation notation is
represented by the Latin word **Ã¥**** **(sigma).

In this article, we will discuss the formula
of summation notation and all terms related to it. We will learn how to write
finite sequences of numbers in summation form. We will discuss how to expand
the sigma notation. To understand summation notation, we solve many examples
related to summation notation.

## Summation formula

By using sigma notation,
we can write the sum of several numbers in a concise form. Mathematically it is
defined as

Ã¥_{i=m}^{n} a_{i} = a_{m} + a_{m+1
}+ a_{m+2} + … + a_{n}

Read as the sum of a_{i} where i
goes from m to n.

Here,

·
**a _{i}** = summands or indexed
variable

·
**i** = index of summation

·
**m** = lower bound of summation

·
**n **= upper bound of summation

It looks scary however, it works very
amazingly. It closes many finite numbers in just one symbol. If we want to
write the sum of the first hundred natural numbers, then we can write Ã¥_{n=1}^{100} n instead of 1 + 2 + 3 + 4 + … +
100.

**Note:** The sum of many numbers that do not follow any specific pattern
can’t be written in form of sigma notation.

## Method to write summation notation

Let’s learn how to write sigma notation
with finite numbers that follow a specific pattern

·
Use any English lowercase
letter to write the index of summation.

·
Observe the pattern of the
given sequence and make a formula for the sequence as given in follows,

The sequence of the numbers |
Formula by using n index |

Natural number |
n |

Even |
2n |

odd |
2n + 1 |

Square |
n |

Multiple of k, for all k = 1,2,3…n |
k n |

·
Create upper and lower bounds
of summation according to the pattern.

·
Use the sigma symbol and write
the first number of the given sequence as a lower bound and the last number
will be used as an upper bound summation.

·
For more understanding, we will
solve some examples in the example section.

## Method to break(Expand) the sigma notation

Expanding of sigma notation is the inverse
of writing the sigma notation. Expand the sigma notation by the following
steps.

·
Observe the indexed variable
and lower, and upper bound that is used with sigma notation.

·
Put the values one by one in the
given indexed variable from the lower to the upper bound of summation.

·
Also, put the plus sign between
each obtained term.

## Properties of sigma
notation

**
i.
****Commutative properties concerning addition and subtraction **

Ã¥ _{i=m}^{n} (a_{i} ± b_{i}) = Ã¥ _{i=m}^{n} a_{i} ± Ã¥ _{i=m}^{n}
b_{i}

**
ii.
****Quotient and product properties do not hold in sigma notation i.e. **

Ã¥ _{i=m}^{n} (a_{i} × b_{i}) ≠ Ã¥ _{i=m}^{n}
a_{i} × Ã¥ _{i=m}^{n}
b_{i}

Ã¥ _{i=m}^{n} (a_{i} / b_{i}) ≠ Ã¥ _{i=m}^{n}
a_{i} / Ã¥ _{i=m}^{n}
b_{i}

**
iii.
****Summation notation for any constant**

Ã¥ _{i=m}^{n} (k × a_{i}) = k Ã¥ _{i=m}^{n} (a_{i})

Where k is any constant.

**
iv.
****Break the sigma notation**

Ã¥ _{i=m}^{n} (a_{i}) = a_{m} + Ã¥ _{i=m+1}^{n}
(a_{i})

A sigma calculator
is an online resource that is helpful for finding the summation of the function
according to the properties of the sigma notation.

## Example of sigma notation

To understand how to sigma notation and how
to expand it, we will solve some examples for you.

**Example.**

Write the following sequence into sigma
notation

1 + 3 + 5 + 7 + 9 + 11 … + 49

**Solution.
**

Step 1. Use any index say i. Since it is an odd number, we know that odd
numbers are written as 2i + 1. So,

index variable = 2i + 1

Step 2. Create the lower and upper bound of summation according to 2i + 1.
If we put 0 in it, we get the first term of a given sequence. Look at 2i + 1,
where 1 is added and 2 multiplied with i. We calculate the inverse of it to get
the upper bound of summation. Subtract 1 from 49, divide it by 2, and get 24.

So,

The lower bound of summation = m = 0

The upper bound of summation = n = 24

Step 3. By using the definition of sigma notation, we can write the above
sequence in sigma notation as

Ã¥ _{i=1}^{24} (2i + 1)

It read as the sum of (2i + 1) from i = 0
to 24.

**Example
2.**

Expand the following sigma notation, also
simply it.

Ã¥ _{i=1}^{5} (2i^{2})

**Solution.
**

Where,

The upper bound
of summation = 1

The lower bound
of summation = 5

Put these values
and all values between them in 2i^{2}

2(1)^{2}, 2(2)^{2}, 2(3)^{2},
2(4)^{2}, 2(5)^{2}

Write a positive sign between them, and we
get

Ã¥ _{i=1}^{5}
(2i^{2}) = 2(1)^{2} + 2(2)^{2} + 2(3)^{2} + 2(4)^{2}
+ 2(5)^{2}

Ã¥ _{i=1}^{5} (2i^{2}) =2 Ã¥ _{i=1}^{5}
(i^{2}) = 2[ (1)^{2} + (2)^{2} + (3)^{2} + (4)^{2}
+ (5)^{2}]

Now simplify it

Ã¥ _{i=1}^{5} (2i^{2}) = 2[ 1 + 4 + 9 + 16 +
25]

= 2(55)

Ã¥ _{i=1}^{5} (2i^{2}) = 110

## Conclusion

In this article, we have discussed the
summation(sigma) notation with its formula. Then we learn how to change
sequence into summation notation. Also, we discussed the method of expanding
the sigma notation. We covered important properties of summation notation.

Further, In the examples section, we solved
some examples of writing sequences into sigma and expanding the sigma notation.
After reading this article, you will be able to solve any problems related to
the sigma notion.

## No comments