A compact and practical way to describe the summation of a series of items is to use summation notation, often known as sigma notation. Sigma (∑) is the Greek number represent the summation notation followed by an expression, with an index variable specifying the range of values to be summed.

In this article, we will explore the definition and concept of summation notation, including the steps for expansion, key formulas, and its practical applications. 

Summation Notation

Summation notation, denoted by ∑, succinctly represents the addition of a sequence of terms over a specified range using an index variable. Summation notation simplifies complex calculations and helps mathematicians and scientist’s express mathematical concepts more elegantly.

Summation notation is a method for representing a very big number in summation form in mathematics. When a series is large, in that case we use the summation form for our better understanding. Summation notation allows for a concise representation of addition operations over a specified range using an index variable.

Summation notation is represented as given below.

Summation Notation

Steps used to expand summation notation    

Expanding summation notation involves expressing the series of terms explicitly by listing and calculating each term individually. 

Understand the Notation: First, make sure you understand the given summation notation. It typically consists of ∑ (sigma), an index variable (usually represented by a letter like i or k).

Find Range: Determine the range of values that the index variable will take, starting from the lower limit to the upper limit. 

Write Out the Terms: Write out the expression inside the summation for each value of the index variable within the specified range. Replace the index variable with its respective value. 

Calculate the Sum: Add up all the individual terms from step 3 to find the final result. 

Formulas of summation notation 

NameFormulaExpand form
First n natural number sum.ni=1 xi =1+2+3+…+n=n(n+1)/2
Sum of squareni=1 i2 =12+22+32+…+n2=n(n+1)(2n+1)/6
Sum of cubeni=1 i3 =13+23+33+…+n3=n2(n+1)2/4
First n odd number sum.ni=1 (2i+1)=1+3+5+…+(n numbers)= n2
Arithmetic sequence ni=1 a+(i-1)d=n/2[2a+(n-1)d]

Use of summation notation 

Summation notation, denoted by the symbol ∑ (sigma), is a powerful mathematical tool with various practical uses in different fields of mathematics and science. Here are some common applications of summation notation:

Summation of Series: Summation notation is used to express and calculate the sum of a series of numbers or terms. For example, it’s employed in calculus to represent and evaluate infinite series like geometric series or power series.

Mean Calculations and Average: In statistics, to find the average or mean we use summation notation. Let suppose, the mean of a set of n data points can be represented as ∑xi /n, where xi represents each data point.

Discrete Probability: In probability theory, summation notation is used to calculate probabilities in discrete random variables. 

Riemann Sums: In calculus, summation notation is used to define Riemann sums, which are used to approximate definite integrals. This concept is fundamental for understanding integral calculus.

Matrix Operations: In linear algebra, summation notation is employed to perform matrix operations. For instance, multiplying two matrices involves summation notation to compute each element of the resulting matrix.

Recursive Definitions: Summation notation can describe recursive mathematical relationships, such as the Fibonacci sequence, where each term depends on the sum of previous terms.

Computer Science and Algorithms: Summation notation is useful in computer science to analyze algorithms, particularly when calculating time complexity. 

Engineering and Physics: Engineers and physicists often use summation notation to express complex physical phenomena, such as the summation of forces in mechanical systems or the summation of charges in electrical circuits.

Finance: In finance and economics, summation notation can represent the present value or future value of a series of cash flows, helping to calculate the net present value (NPV) and other financial metrics.

Statistical Analysis: In regression analysis, summation notation is used to define the formulas for estimating coefficients in linear regression models.

Examples of Summation Notation 

Example number 1:

Find the sum of first 10 whole numbers.


Step 1: To find the sum write all numbers and place plus sign between each term.

0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

Step 2: Add 

Sum of first 10 whole numbers = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

Example number 2:

By using the summation formula determine the value of ∑20i=0 (m + 3)3.


Step 1: Given data 

20i=0 (m + 3)3.

(0 + 3)327
(1 + 3)364
(2 + 3)3125
(3 + 3)3216
(4 + 3)3343
(5 + 3)3512
(6 + 3)3729
(7 + 3)31000
(8 + 3)31331
(9 + 3)31728
(10 + 3)32197
(11 + 3)32744
(12 + 3)33375
(13 + 3)34096
(14 + 3)34913
(15 + 3)35832
(16 + 3)36859
(17 + 3)38000
(18 + 3)39261
(19 + 3)310648
(20 + 3)312167

Total Sum = 27 + 64 + 125 + 216 + 343 + 512 + 729 + 1000 + 1331 + 1728 + 2197 + 2744 + 3375 + 4096 + 4913 + 5832 + 6859 + 8000 + 9261 + 10648 + 12167

Total Sum = 76167

You can also try a sigma calculator to avoid such a lengthy calculations of expanding summation notation 


In this article, we have discussed the definition, concept, steps used to expand summation, formulas, and use of summation notation. Moreover, for a better understanding of the reader summation notation is more explained with the help of detailed examples. Anyone can defend this article after studying this article.


Question Number. 1:

How does summation notation work?


Summation notation consists of Σ followed by an index variable, a lower limit, an upper limit, and an expression. It calculates the sum of the expression as the index variable varies over the specified range.

Question Number. 2:

What is the purpose of using summation notation?


Summation notation provides a concise and standardized way to express and compute sums of sequences, making it easier to work with mathematical and statistical concepts.

Question Number. 3:

What are some common applications of summation notation?


Summation notation is widely used in calculus, statistics, discrete mathematics, and various scientific and engineering fields to describe and calculate sums, averages, probabilities, and more.