In this era of technology, the use of different products of household things is increasing on a wider scale. The summation is widely used in solving different educational and daily life problems. Such as, when a person went to a grocery shop to buy several things.

Then the total price must be the sum of different prices of the products that are bought. For this purpose, summation notation is widely used. In this article, we are going to explore all the basics of the summation notion along with examples and solutions.

## Summation Notation – Definition

The summation notation also known as sigma notation is usually a Greek notation used to denote the sum of several numbers by making a function with initial and final indexes. In hypothesis testing, a sum of squares, variance, and many other similar terms, the sigma notation is used.

### Notation of sigma

The term sigma or summation is denoted by the symbol “∑”. The initial term of the sequence or a function must be at the subscript and the final term must be at the superscript of the notation.

**N _{i} = N_{1} + N_{2} + N_{3} + … + N_{n}**

- In the above expression, the “∑” is the notation of summation.
- “N
_{i}” is the group of number or a summation function in which the values from initial to final has to be substituted. - “i = 1” is the initial term of the function.
- “n” is the final term.

## Two ways of finding the sum

Here are two ways of finding the sum of the list of numbers.

- Simple summation
- Sigma notation

### 1. Simple summation

Simple summation is widely used to find the sum of the list of numbers. It is also known as the method of long addition. For example, the sum of the first ten even numbers can be calculated by using the method of simple summation.

Sum of first 10 even numbers = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20

Sum of first 10 even numbers = 110

### 2. Sigma (summation) notation

The other method of finding the sum of the list of numbers is summation notation. In summation notation, the numbers are written in the form of a function with a starting and ending point. The lower index is the starting and the upper index is the ending term of the sequence.

The sequence of the first 10 odd numbers can be written as a summation notation of 2x – 1 with the lower point at 1 and the upper point at 10. Such as

[2x – 1] = (2(1) – 1) + (2(2) – 1) + (2(3) – 1) + (2(4) – 1) + (2(5) – 1) + (2(6) – 1) + (2(7) – 1) + (2(8) – 1) + (2(9) – 1) + (2(10) – 1)

= (2 – 1) + (4 – 1) + (6 – 1) + (8 – 1) + (10 – 1) + (12 – 1) + (14 – 1) + (16 – 1) + (18 – 1) + (20 – 1)

= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

= 100

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## Laws of sigma notation

Law Name | Law |

Constant Law | k = n * k |

Sum Law | P_{x} + Q_{x} = P_{x} + Q_{x} |

Difference Law | P_{x} – Q_{x} = P_{x} – Q_{x} |

Product Law | P_{x} * Q_{x} = P_{x} * Q_{x} |

Quotient Law | P_{x} / Q_{x} = P_{x} / Q_{x} |

## How to calculate the summation?

To calculate the summation there are two ways such as:

- By using a summation calculator
- By using the manual method

Here is the description of using these methods.

### By using a summation calculator

As it is the era of technology, there are many online calculators are available to help students for getting step-by-step solutions to their complex problems. The problems of summation notation can be calculated easily by using a summation calculator.

**How to use this calculator?**

Follow the below steps.

- Select the method such as simple summation or sigma notation.
- Enter the initial value, expression, and final values into the required input fields.
- Select the corresponding variable “x” is selected by default.
- Press the calculate button.
- The solution with steps will come below the calculate button.

### By using the manual method

Here are a few examples of summation to learn how to calculate it manually.

**Example 1**

Find the sum of the given expression if the starting value is 1 and the ending value is 6.

f(x) = 12x^{3} – 15x^{2} + 12

**Solution**

**Step 1:** Write the summation function in the form of summation notation.

(12x^{3} – 15x^{2} + 12)

**Step 2:** Apply the initial and final values directly by putting x = 1, 2, 3, …, 6.

**For x = 1**

12x^{3} – 15x^{2} + 12 = 12(1)^{3} – 15(1)^{2}+ 12

= 12(1) – 15(1)+ 12

= 12 – 15 + 12 = 9

**For x = 2**

12x^{3} – 15x^{2} + 12 = 12(2)^{3} – 15(2)^{2}+ 12

= 12(8) – 15(4)+ 12

= 96 – 60 + 12 = 48

**For x = 3**

12x^{3} – 15x^{2} + 12 = 12(3)^{3} – 15(3)^{2}+ 12

= 12(27) – 15(9)+ 12

= 324 – 135 + 12 = 201

**For x = 4**

12x^{3} – 15x^{2} + 12 = 12(4)^{3} – 15(4)^{2}+ 12

= 12(64) – 15(16)+ 12

= 768 – 240 + 12 = 540

**For x = 5**

12x^{3} – 15x^{2} + 12 = 12(5)^{3} – 15(5)^{2}+ 12

= 12(125) – 15(25)+ 12

= 1500 – 375 + 12 = 1137

**For x = 6**

12x^{3} – 15x^{2} + 12 = 12(6)^{3} – 15(6)^{2}+ 12

= 12(216) – 15(36)+ 12

= 2592 – 540 + 12 = 2064

**Step 3:** Now take the sum of all the above-calculated terms.

(12x^{3} – 15x^{2} + 12) = 9 + 48 + 201 + 540 + 1137 + 2064

= 3999

**Alternately**

**Step 1:** Apply the notation of summation to each function separately by using the sum and difference rule.

(12x^{3} – 15x^{2} + 12) = (12x^{3}) – (15x^{2}) + (12)

**Step 2:** Find the sum of each function separately by putting the value in the function.

**For ****(12x ^{3})**

(12x^{3}) = 12(1)^{3} + 12(2)^{3} + 12(3)^{3} + 12(4)^{3} + 12(5)^{3} + 12(6)^{3}

= 12(1) + 12(8) + 12(27) + 12(64) + 12(125) + 12(216)

= 12 + 96 + 324 + 768 + 1500 + 2592

= 5292

**For ****(15x ^{2})**

(15x^{2}) = 15(1)^{2} + 15(2)^{2} + 15(3)^{2} + 15(4)^{2} + 15(5)^{2} + 15(6)^{2}

= 15(1) + 15(4) + 15(9) + 15(16) + 15(25) + 15(36)

= 15 + 60 + 135 + 240 + 375 + 540

= 1365

**For ****(12)**

(12) = 12 + 12 + 12 + 12 + 12 + 12

= 12 x 6

= 72

**Step 3:** Write the results of the summation functions and find their order of operation of them.

(12x^{3} – 15x^{2} + 12) = (12x^{3}) – (15x^{2}) + (12)

= 5292 – 1365 + 72

= 3927 + 72

= 3999

## Final Words

In this post, we have covered all the basics of the summation notation along with solving it in various ways. Now you can solve any problems with sigma notation by following the above techniques used in the examples.