What are the different methods of summation and where do we use summation?

Summation is the addition of a sequence of any form of numbers, known as addends or summands, with the result being their sum or total in mathematics. Other sorts of values, in addition to numbers, can be summed: functions, vectors, matrices, polynomials, and, in general, members of any form of mathematical object that has the "+" operation defined.

Series are summations of infinite sequences. They are not covered in this article since they entail the concept of limit.

What are the basic methods of the Summation?

Summation use sigma notation which is applied to the functions. The summation of any sequence or series can easily be calculated by using summation calculator. Let’s discuss methods to perform summation on numbers and functions.

Method of sum:

When sigma notation of a function of two numbers having a sum sign is given, we can use this method. According to this method, sigma notation is applied separately to both functions. Such as, if a and b are two functions.

Example :

Find the sum of the given function:

Solution :

Step 1: Write the given function.

Step 2: Apply method of sum.

Step 3: Identify the values of the sigma notation.

X = 0, 1, 2, 3

Step 4: Put the above values of x in the

For x = 0

2x = 2(0) = 0

For x = 1

2x = 2(1) = 2

For x = 2

2x = 2(2) = 4

For x = 3

2x = 2(3) = 6

Step 5: Put the above values of x in the

For x = 0

3x = 3(0) = 0

For x = 1

3x = 3(1) = 3

For x = 2

3x = 3(2) = 6

For x = 3

3x = 3(3) = 9

Step 6: Put the values in the equation.

Method of Difference:

When sigma notation of a function of two numbers having a subtraction sign is given, we can use this method. According to this method, sigma notation is applied separately to both functions. Such as, if a and b are two functions.

 ∑(x=0)^n (ax - bx) =  ∑(x=0)^n (ax) - ∑(x=0)^n (bx)

Example:

Find the difference of the given function ∑(x=0)^3 (4x - 3x).

Solution:    

Step 1: Write the given function.

∑(x=0)^3 (4x - 3x)                                        Note: ∑(x=0)^3 means, ∑ limit from 0 to 3

Step 2: Apply the method of difference.

∑(x=0)^3 (4x - 3x) = (4x) -(3x)

Step 3: Identify the values of the sigma notation.

X = 0, 1, 2, 3

Step 4: Put the above values of x in the ∑(x=0)^3 (4x).

For x = 0

4x = 4(0) = 0

For x = 1

4x = 4(1) = 4

For x = 2

4x = 4(2) = 8

For x = 3

4x = 4(3) = 12

Step 5: Put the above values of x in the ∑(x=0)^3 (3x).

For x = 0

3x = 3(0) = 0

For x = 1

3x = 3(1) = 3

For x = 2

3x = 3(2) = 6

For x = 3

3x = 3(3) = 9

Step 6: Put the values in the equation.

∑(x=0)^3 (4x - 3x) = (0 + 4 + 8 + 12) - (0 + 3 + 6 + 9)

                           = 24- 18

                           = 6

Method of constant:

When sigma notation of a constant function is given, we can use this method. According to this method, when sigma notation is applied to that constant it remains unchanged. Such as, if a is any constant.

∑(x=0)^n(a) = a

Example

Find the given function (4).

Solution

Step 1: Write the given function.

∑(x=0)^3 (4)         Note: ∑(x=0)^3 means, ∑ limit from 0 to 3

Step 2: Apply the method of constant.

∑(x=0)^3 (4) = 4

Why do we use summation and how to calculate it?

 Summation or sigma notation can be used to represent a series in a concise format. Many variables are frequently required in mathematical formulas. Summation, often known as sigma notation, is a shorthand method for expressing the sum of a variable's values concisely.It is more vital in the life of sciences to be able to understand a summation notation that has been supplied to you than it is to be able to express a given sum in summation notation. A sum of numbers is represented using summation notation.

The additive impact of multiple electrical impulses on a neuromuscular junction, the intersection between a nerve cell and a muscle cell, is known as summation in physiology. Individually, the stimuli are unable to elicit a reaction, but when combined, they can.

A sum of numerous terms is usually denoted by the symbol Σ(sigma). This symbol is usually followed by a variable index that includes all terms that must be included in the sum. The sum of the first whole numbers, for example, can be represented as follows: 0 + 1 + 2 + 3 + …

Let’s take an example.

Example

Find the summation of the given function ∑(x=0)^7 (4x² - 3).

Solution:

Step 1: Write the given function.            Note: ∑(x=0)^7 means, ∑ limit from 0 to 7

∑(x=0)^7 (4x² - 3)

Step 2: Identify the values of the sigma notation.

X = 0, 1, 2, 3, 4, 5, 6, 7

Step 3: Put the above values of x in the ∑(x=0)^7 (4x² - 3).

For x = 0

4x² - 3 = 4(0)² - 3 = 0 – 3 = -3

For x = 1

4x² - 3 = 4(1)² - 3 = 4(1) – 3 = 4 – 3 = 1

For x = 2

4x² - 3 = 4(2)² - 3 = 4(4) – 3 = 16 – 3 = 13

For x = 3

4x² - 3 = 4(3)² - 3 = 4(9) – 3 = 36 – 3 = 33

For x = 4

4x² - 3 = 4(4)² - 3 = 4(16) – 3 = 64 – 3 = 61

For x = 5

4x² - 3 = 4(5)² - 3 = 4(25) – 3 = 100 – 3 = 97

For x = 6

4x² - 3 = 4(6)² - 3 = 4(36) – 3 = 144 – 3 = 141

For x = 7

4x² - 3 = 4(7)² - 3 = 4(49) – 3 = 196 – 3 = 193

Step 4: Sum all the results.

-3 + 1 + 13 + 33 + 61 + 97 + 141 + 193 = 536

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