In mathematics there are many types of arithmetic operations, one of which is addition. Of course it’s very easy if you just count 1+2+3+4+5+6, and it can also be done manually. But what if the series of numbers reaches hundreds or thousands? In this case sigma notation is needed.

You can imagine that if you had to manually type a sum with a series of numbers up to hundreds or thousands, it would definitely be very long and confusing. To understand more about this material, please see the following discussion.

## Understanding sigma notation

Before discussing this material further, we should know that sigma itself is denoted by the symbol âˆ‘. This symbol may not be familiar to many people, and quite a few people probably read it as the letter E, even though it is sigma.

### The origin of the symbol âˆ‘

It is known that the sigma symbol originates from the 18th letter of the Greek alphabet and is read with the letter S in Latin. Meanwhile, the Greeks used it as a symbol for SUM or adding a series of numbers.

In mathematics and natural sciences, the symbol âˆ‘ or sigma is used to indicate addition. For your information, the symbol âˆ‘ was first used by the Swiss mathematician Leonhard Euler in 1755.

### Definition âˆ‘

This symbol was chosen because it illustrates the term SUM, which begins with the letter S and has the symbol âˆ‘, or sigma in Greek. If we return to the function of this symbol, we can conclude that what is meant by the sigma symbol is:

A form of writing intended to summarize the sum of terms into a series of numbers. The terms in addition follow a pattern; they cannot be a random pattern.

This material is also closely related to the subjects of series and sequences in geometry and arithmetic. Therefore, to facilitate understanding, you must have mastered the material related to lines and series in geometry and arithmetic.

## sigma notation formula

To perform addition operations using sigma, you need a useful basic formula for these calculations. In fact, the basic formula for addition is very simple, but unfortunately many people do not understand it, so they are deceived and get the wrong answer.

In general, the formula used in this calculation is:

Information:

âˆ‘ is sigma notation

*Sh*i is the formula or fourth term*I*

*I *It is the sum index

*s *It is the minimum sum index

*n *It is the upper limit of the sum index

More complete information is as follows:

- The upper limit is the last number in the addition that must be reached and the lower limit is the first number in the addition that is used to start the calculation.

- Formula or quarter
*I*It is the equation that will be used in each indicator, whether the lower limit indicator or the upper limit indicator. Meanwhile, the sum index is an index that will later be included in the formula as a variable.

To make sigma calculation easier, you also need to reunderstand the material on FPB and KPK that you studied in basic mathematics.

In this case, the sigma notation must also refer to the general formulas present, as in the following explanation:

Based on the general formula above, it is clear that:

*user interface *= 2*i + 5*

This means you need to add all the terms, i.e. (2*I *+ 5) l *I *= 1 l *I *=5. So the calculation is:

= (2(1) + 5) + (2(2) + 5) + (2(3) + 5) + (2(4) + 5) + (2(5) + 5)

= 7 + 9 + 11 +13 + 15

= 55

## Properties of sigma notation

To understand more about this mathematics topic, you not only need to memorize basic formulas, but you also need to understand the properties of these formulas. So that you can later apply the most appropriate formula in your calculations.

For greater clarity, the following will explain in detail the characteristics of sigma notation calculations:

- The first property appears in the following form:

The above property shows that in sum, the term has a value of 1 in the index *I *= 1 l *n *Then it will produce a number *n *Itself. Examples are as follows:

In this formula, it becomes clear that the nature of the sigma notation indicates the existence of a constant such that it does not need to be included in the addition process directly.

Later, the constants in front of the terms can be multiplied after the final addition result is found. Here is an example:

= 3(2^{2 }+ 3^{2 }+ 4^{2 }+ 5^{2})

= 3 (4 + 9 + 16 + 25)

= 3 (54)

= 162

- The third characteristic is as follows:

The above properties of sigma notation apply to the addition of two different terms. If you encounter a problem like the one above, you need to find the sum of each term.

- The fourth characteristic is:

This property shows that the above two sigmas have formulas with the same terms but different terms. The condition is that the lower bound of the second sigma must be n+1 or be a continuation of the upper bound of the first sigma.

In this way, the new sigma will have the same lower bound as the first sigma, while the upper bound will be the same as the second sigma. Here is an example:

And others.

## Further clarifications about the properties of sigma notation

Please note that when studying material about sigma, you need to know what properties it contains. Such as the index can be changed or the notation can be changed. So that you don’t get confused in this discussion:

### 1. Index typing can be changed

Basically, writing the index in sigma doesn’t need to use the letter i, you can just change it to the letter a, k, or l. However, you must ensure that if the index letter is changed, the index letter must also be written in the same way in the mathematical sentence.

In addition, the letter that is used to change the index cannot be the same letter that will be used to indicate the upper limit of sigma. Here is an example:

### 2. The form of sigma notation can be changed

Not only the index can be changed, in sigma notation you can also change its form by separating it into two or more additions. Here are some examples:

- Divided into two or more groups

From the above example it can be seen that the lower bound of the indicator is 1 while the upper bound is n. They can then be separated into two sigma parts and added together. The first is to have a lower bound of 1 and an upper bound of m.

Meanwhile, the second has another lower bound in the form of m+1 and an upper bound in the form of n. Therefore, the index boundaries and the boundaries between the first and second must be sequential. Likewise if it is divided into three increments.

An example question is as follows:

You can pay attention to the sigma notation above and we will translate it into a sum as follows:

Once this is explained, it can then be divided into two groups. They are one sigma with an index from 2 to 7, and the second is a sigma with an index from 8 to 12. The explanation is as follows:

You can also separate the first and last terms

This property is very different from the separation described previously, because in sigma notation what is separated is the last term or *And the. *For more details, please see the example calculation below:

You can first focus on the left side, which if explained would look like this:

From the above explanation, it is clear that the last term in the addition is (2n + 3). If you have understood up to this point, we can now enter the last term in the form of an addition. The result will be like this:

## Example of sigma notation questions

To make it easier to understand the discussion about the material this time, you can immediately listen to several example questions below:

**1. Calculate the result of the addition below**

The question above asks you to add everything from i=5 to 15. Below is a discussion of how to do this.

a favour:

n or the upper limit is 15

I or minimum is 5

The formula is I

Answer:

= 110

**2. What is the result of the following addition?**

In this question, you are asked to find the total number of all (7 – i) from the lower bound, which is 1, to the upper bound, which is 6. Below is the discussion.

a favour:

n or upper limit = 6

I or minimum index = 1

Formula = 7 – i

Answer:

**3. Determine the results of the following sigma notation:**

In the question above, you are asked to add all (i+1) starting from the lower bound, which is i=-2, and the upper bound, which is 2. Here is the discussion:

a favour:

n or upper limit sigma = 2

i or min = -2

Formula = i + 1

Answer:

= â€“ 1 + 0 + 1 + 2 + 3

= 5

**4. What is the result of the following addition:**

Based on this question, it can be explained as follows:

This can be explained further as follows:

= 3(1^{2 }+ 2^{2} + 3^{2 }+ 4^{2}) + 4 (1 + 2 + 3 + 4)

= 3 (30) + 4 (10)

= 310

Basically, material related to sigma notation must be well understood because it is often used to support other material such as series, sequences, and mathematical induction. Moreover, the main function of this calculation is to summarize the addition process so that it is not too long.