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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 å (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

åi=mn ai = am + am+1 + am+2 + … + an

Read as the sum of ai where i goes from m to n.


·         ai = 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 ån=1100 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





2n + 1



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

å i=mn (ai ± bi) = å i=mn ai ± å i=mn bi

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

å i=mn (ai × bi) å i=mn ai × å i=mn bi

å i=mn (ai / bi) å i=mn ai /  å i=mn bi

    iii.            Summation notation for any constant

å i=mn (k × ai) = k å i=mn (ai)

Where k is any constant.

    iv.            Break the sigma notation

å i=mn (ai) = am + å i=m+1n (ai)

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.


Write the following sequence into sigma notation

1 + 3 + 5 + 7 + 9 + 11 … + 49


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.


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 

å i=124 (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.

å i=15 (2i2)



The upper bound of summation = 1

The lower bound of summation = 5

Put these values and all values between them in 2i2

2(1)2, 2(2)2, 2(3)2, 2(4)2, 2(5)2

Write a positive sign between them, and we get

 å i=15 (2i2) = 2(1)2 + 2(2)2 + 2(3)2 + 2(4)2 + 2(5)2

å i=15 (2i2) =2 å i=15 (i2) = 2[ (1)2 + (2)2 + (3)2 + (4)2 + (5)2]

Now simplify it

å i=15 (2i2) = 2[ 1 + 4 + 9 + 16 + 25]

                   = 2(55)

å i=15 (2i2) = 110


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.

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