What is the summation notation? Its Properties and Examples

Summation notation is also known as sigma notation in mathematics, shorthand is used to represent the sum of a sequence of numbers. The notation was first introduced by the Swiss mathematician Leonhard Euler in the 18th century and later popularized by Carl Friedrich Gauss.

The notation uses the symbol “∑” which is the Greek letter sigma that represents the sum of a sequence of terms. The sequence is usually defined by an index variable, such as (i or n), and a lower and upper limit.

sumation notation

Another benefit of summation notation is that it enables mathematicians to express recursive formulas and algorithms concisely and elegantly. This is particularly useful in computer programming and other fields where iterative processes are used to solve complex problems.

Summation notation is widely used in various fields of mathematics, including calculus, statistics, and number theory, as well as in physics and engineering.

In this article, we will discuss the basic definition, formulas, properties, and how to find them.

Summation Notation

The most common technique of writing infinite numbers of terms in a sequence is called summation notation (a.k.a sigma notation). While using summation notation, the variable written below the sigma is called the index of summation. The lower number denotes the lower limit of the index (it denotes the summation begins) whereas the upper number denotes the upper limit of summation (it denotes the summation ends).


The symbol can be represented in the Greek letter sigma “∑”. When the sigma symbol is used in a mathematical expression. The symbol of sigma is (∑ni=1 xi).

  • Firstly,” i“ is the index of the summation.
  • “1“is the initial value of summation or the lower limit of summation.
  • “n” is known as the ending point or upper limit of the summation.
  • The symbol “∑” is called summation.
  • Variable “xi” is equal to the element of summation.

Working methodology of summation notation

Here are some techniques to calculate the summation notation given below:

  • Identify the index variable and the lower and upper limits of the summation notation.
  • Write out the sequence of terms defined by the index variable and the limits.
  • Add up the terms in the sequence to get the sum.
  • If the sequence has a pattern or formula, use it to simplify the sum.
  • Replace the index variable with its corresponding values from the limits to evaluate the sum.
  • Add up the resulting values to get the final sum.
  • If the limits are variables or expressions, simplify them before evaluating the sum.
  • If there are multiple summation notations, evaluate them one at a time, starting from the innermost one.
  • If the summation notation has additional terms or coefficients, distribute them before evaluating the sum.
  • Check your answer to make sure it is correct and matches the original problem.

Summation notation Formulas

Here important formulas of summation are given below:

  • mi=1 k xi = k ∑mi=1 xi
  • mi=1 (xi ± yi) = ∑mi=1 xi + ∑mi=1 yi
  • mi=1 k = km
  • mi=1 I =[m(m + 1)]/2
  • mi=1 I2 =[m(m + 1)(2m + 1)]/6
  • mi=1 I3 =[{m(m + 1)}/2]2

Properties of the Summation Notation:

Summation notation represented using the symbol ∑ (sigma), has several important properties that are used in mathematics. Some of the important properties of the summation notation are defined below.


The sum of two or more sequences can be written as the sum of each sequence separately. In other words, for any constants L and M, and any sequences xi and yi, we have:

mi=1 (L + M (yi)) = L mi=1(xi) + M mi=1(yi)


The order of the terms in a sum can be changed without disturbing the results. In other words, for any sequence xi and yi, we have:

mi=1 (xi + yi) = mi=1 (yi + xi)


The combination of terms in a sum can be changed lacking moving the result. There are the sequence xi, yi, and zi, we have:

mi=1 {xi + (yi + zi)} = mi=1 {(xi + yi)+ zi}


A constant inside the summation is taken out from the summation. In other words, for any constant k and any sequence xi, we have:

mi=1 (k x) = k mi=1 (xi)

Swap Limits of Summation:

It will not affect the results if the limits of a summation change. In other words, for any sequence xi and any integer’s m and n such that m = n, we have:

mi=1 (xi)= nj=1 (xi)

Where “j” is the index “xj”.

These properties of summation notation are important in mathematical analysis and help simplify many calculations involving sums.

How to solve summation notation problems?

Follow the below examples to understand how to evaluate summation problems.

Example 1:

Find the sum of the first 12 even numbers.


Step 1:

First, we write the given number and its index value.

Let x be a variable used for the sum of numbers.

Start value= 1

Ending value = m = 12

Step 2:

Also, we write the summation formula in terms of the taken variable.

mx=1 (x)

Step 3:

Put the above value in the summation formula.

12x=1 (x)

Step 4:

Open the summation by definition.

12x=1 (x) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12

12x=1 (x) = 78

Step 5:

Hence, ∑12x=1 (x) = 78, which is the sum of the first 12 even numbers.

Example 2:

Determine the given summation that is ∑8x=1 (15x+ 10).


Step 1:

First, we write the given summation carefully.

8x=1 (15x+ 10)

Step 2:

Now, with the help of the linearity property evaluate the above summation.

Linearity Formula

 mi=1 (L xi + M yi) = L mi=1(xi) + M mi=1(yi)

Step 3:

Apply the Linearity Formula and simplify.

8x=1 (15x+ 10)

= 15 ∑8x=1 (x) + ∑8x=110)

Step 4:

8x=1 (15x+ 10)

= {15 (1) + 10} + {15(2) + 10} + {15(3) + 10} + {15(4) + 10} + {15(5) + 10} + {15(6) + 10} + {15(7) + 10} + {15(8) + 10}

= {15 + 10} + {30 + 10} + {45 + 10} + {60 + 10} + {75 + 10} + {90 + 10} + {105 + 10} + {120 + 10}

= {25 + 40 + 55 + 70 + 85 + 100 + 115 + 130}

= 620

Therefore, ∑8x=1 (15x+ 10)= 620, is the solution of∑8x=1 (15x+ 10).

A summation notation calculator can also be used to evaluate the sigma notation problems quickly.

Frequently asked questions

Question 1:

Define Summation Notation


Summation notation is a concise way of representing a sum of a sequence of numbers. It is called sigma notation, where the Greek letter sigma ∑ represents the sum. The notation consists of the symbol ∑ followed by the terms to be summed, enclosed in parentheses, and separated by commas. The general form of the summation notation is ∑nx=1 (xi).

Question 2:

What does the mathematical symbol ∑ mean?


In mathematics, the symbol ∑ denotes the summation operator or sigma notation. It is utilized to signify the total of a series of terms.


In this article, we have discussed the basic definition of summation notation, how to calculate the summation notation with formulas, and the properties of summation notation in detail. And also with the help of an example topic will be explained. After complete studying this article anyone can easily define this topic.

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