Summation Notation: Definition, Formula, Rules, & Examples

When dealing with mathematics and its various branches, simplification and efficiency are key. Summation notation is an important notation that plays a key role in simplifying the complex and complicated expressions of series, etc. Summation notation (Also known as sigma notation) provides a precise and concise way to write the summation of a series of numbers or terms.

Summation notation is an essential tool in mathematics and its many applications because of how efficiently and elegantly it represents sums. Mathematical expressions are easier to acquire and comprehend since they make managing complicated sequences and series simpler.

In this article, we will describe a significant notation summation. We will explore its definition, formula, and some useful rules. We will also give some examples to apprehend significant calculations for series.

What is Summation Notation?

Summation notation is a concise technique for presenting the sum of a series of numbers or terms. It is commonly referred to as sigma notation. It offers a useful shortcut for expressing mathematical series and employs the Greek symbol sigma (Σ) to signify the sums.

The basic structure of summation notation consists of the sigma symbol followed by an expression that specifies the terms to be summed, along with the range over which the summation occurs. Here’s the general form of summation notation:

Σⁿ_{m = i} a_i

In this notation:

• Σ represents the summation symbol.
• a_i represents the individual terms in the sequence.
• i is the index variable that iterates over the sequence.
• m and n define the starting and ending values of the index variable i.

Rules for Using Summation Notation:

Summation notation comes with a set of rules that govern its use and manipulation. These rules help simplify expressions and make mathematical calculations more manageable. Here are some important rules for using summation notation:

Distributive Property:

The distributive property allows you to distribute a constant factor outside the summation:

Σⁿ_{i = m} (c x a_i) = c [Σⁿ_{i = m} a_i]

Sum or Difference:

Σⁿ_{i = m} (a_i ± b_i) = (Σⁿ_{i = m} a_i) ± (Σⁿ_{i = m} b_i)

This rule is useful in dealing with different indices.

Multiplication Rule:

Σⁿ_{i = m} (a_i x b_i) = (Σⁿ_{i = m} a_i) (Σⁿ_{i = m} b_i)

This rule simplifies expressions by merging multiple summations.

Division Rule:

Σⁿ_{i = m} (a_i / b_i) = (Σⁿ_{i = m} a_i) / (Σⁿ_{i = m} b_i)

Changing the Index:

Σⁿ_{i = m} a_i = Σ^q_{j = p} b_j

This rule is employed when you aspire to change the index variable as a summation of the change in the range.

Here, j is a new index unknown, and g_j signifies the relationship between the index variable i and j.

Commutative Property:

Σⁿ_{i = m} a_i + Σ^k_{i = n+1} b_i = Σ^k_{i = m} (a_i + b_i)

This rule is employed to change the order of summation and you can change that sort of order of terms as required in the summation.

These rules make summation notation a versatile tool for simplifying and manipulating mathematical expressions involving series and sequences.

How to expand summation notation?

Expanding summation notation (Σ) involves expressing the compact Σ notation as a series of individual terms. Below is an example to understand it.

Example 1:

Determine the sum of the 1^st 9 prime numbers.

Solution:

Step 1: Given data

1^st 9 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23

Step 2: Now, sum all these given numbers (1^st 9 prime numbers). we will get:

2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100 Ans.

Example 2:

Solve the following series summation.

Σ⁷_{x = 1} (x² + 2x + 2)

Solution:

Step 1: Given data

Σ⁷_{x = 1} (x² + 2x + 2) ………….. (i)

Step 2: Now put the respective values for x in the given expression (x² + 2x + 2) to find the value of the function.

For x = 1

(x² + 2x + 2)= (1)² + 2 (1) + 2

(x² + 2x + 2)= 1 + 2 + 2

(x² + 2x + 2)= 5

For x = 2

(x² + 2x + 2) = (2)² + 2 (2) + 2

(x² + 2x + 2) = 4 + 4 + 2

(x² + 2x + 2) = 10

For x = 3

(x² + 2x + 2)= (3)² + 2 (3) + 2

(x² + 2x + 2) = 9 + 6 + 2

(x² + 2x + 2) = 17

For x = 4

(x² + 2x + 2)= (4)² + 2 (4) + 2

(x² + 2x + 2) = 16 + 8 + 2

(x² + 2x + 2)= 26

For x = 5

(x² + 2x + 2)= (5)² + 2 (5) + 2

(x² + 2x + 2)= 25 + 10 + 2

(x² + 2x + 2)= 37

For x = 6

(x² + 2x + 2) = (6)² + 2 (6) + 2

(x² + 2x + 2) = 36 + 12 + 2

(x² + 2x + 1)= 50

For x = 7

(x² + 2x + 2)= (7)² + 2 (7) + 2

(x² + 2x + 2) = 49 + 14 + 2

(x² + 2x + 1) = 65

Step 3: Put all the values in (i), we have

Σ⁷_{x = 1} (x² + 2x + 2) = 5 + 10 + 17 + 26 + 37 + 50 + 65

Σ⁷_{x = 1} (x² + 2x + 2) = 210 Ans.

You can use a summation calculator to get rid of the above time-consuming and lengthy calculations.

Wrap Up:

In this article, we have addressed an important mathematical notation. We have elaborated on its definition, formula, and significant rules that will assist in solving complex mathematical expressions in a precise and concise way.

In the last section, we’ve solved some examples. Hopefully, by reading and apprehending this article, you will be able to tackle the complex summation problems easily.