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Class 11 Sigma Notation – Definition, Formulas, and Example Problems

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Immaku.com – Have you ever found it difficult to calculate the number of rows of numbers in a mathematical formula? If so, maybe you need to learn sigma notation to simplify these numbers. To make it easy to understand, here is an explanation.

What is Sigma Notation?

what is sigma notation

This basic mathematical material is also called addition notation. It is very important for you to know to be able to learn the basics of advanced mathematics such as calculus and statistics. to get to know him more deeply following his review.

Definition

Sigma notation is simply defined as addition. This formula is used to abbreviate a series of number sums so that they are not too long. This material is still closely related to arithmetic and geometric series.

in science education This math material is used as a sum for a sequence of numbers. Where the result is the sum of these sums. Not only numbers, other forms that can be added are vectors, functions, matrices, and polynomials.

The adder starts from an explicit sequence and is denoted as a succession of additions. There is an associative and communicative addition so that it does not require brackets and the amount does not depend on the order of the peaks.

This sum starts from the sequence of one element that produces the element itself. For example, if the sum starts with a sequence of elements 0, then the conversion is in 0. Meanwhile, for the simple pattern of adding a long series, it is replaced by an ellipse.

An example of adding up 100 natural numbers can be written as 1 + 2 + 3 + 4 + …. + 100. To simplify the formula, use the sigma formula with the symbol Σ. This symbol is a Greek letter called “sum” which means addition.

The formula for this addition notation is written in the following form:

The sum of the first n natural integers is denoted by i-1n1

Sigma Capital Notation Formula

sigma notation

With the same concept, the capital sigma notation uses the symbol above to represent the sum of many terms or a series of numbers. The formula is written as i-mnai = am+ am+i + + an – 1 + an.

In the formula above, i is the sum index. Whereas ai is the index variable for each term of the added amount. The m is the symbol for the lower limit of the sum and n is the upper limit of the sum.

Whereas i=m is a sum symbol indicating the index starts with m, and i is the increment for each term and stops with i=n. Here is an example for adding square numbers.

i=24i2= 22 + 32 + 42 = 29

Sometimes in informal writing the definition of the index and the sum limit are omitted. This is true when it is clear in context. For example in the following formula: ai= i=mnai. In addition there is also a generalization of the notation where arbitrary conditions are provided.

Here are some of the most frequently encountered examples:

  • 0k as the specified range.
  • x∈Sfx is the sum for f(x) which contains all members of x in the set S
  • dln(d) is the sum for (d)for the positive number d divided by n.

Sigma Notation Formula Changes

Do you know? that addition notation has various forms. Starting from changes in the index, as well as writing the notation. The following is clearer.

1. Index changes

What’s interesting about the addition notation is that the index can be changed as needed. So that in writing the index does not have to always use the letter i. You can use other letters such as a, l, or k.

However, when changing it, make sure the index in mathematics is also changed with the same letter. While the letter used to express the upper limit of the sum notation is not the same as the replacement letter in the sigma index. Examples are as follows:

i=1ni = a=1na

From the form of the formula above, we can find out the index in the previous formula using the letter i. Then it is replaced with the letter “a” with a note that the mathematical sentence in the formula uses the same letter according to the index.

Even though the letters have changed, the two sigma formulas still have the same value and don’t change. So that in the formula above using the symbol “equal to” to state that the two sigma formulas are the same.

2. Notation changes

The notation on the sigma formula can also be changed just like the index. To change the notation on sigma, there are two ways you can do it. The first is separated into the sum of two or more sigma notation and the second is by separating the first and last terms.

  • Separated into Sum

The notation on Sigma can be separated or broken down into a number of sums of several sigma formulas. To find out, here is an example of the first notation change:

i=1nui = i=1kui=k+1nui

From the example above, we can break down the formula into two notations. from the first sigma formula the upper limit is the letter n with the index being 1. Then the second formula produces the index k+1, with the index values ​​having to be sequential if you want to add them.

So that when it is broken down into more than two sums, the index values ​​and notations are added sequentially. This is because all values ​​in the upper and lower limit ranges must be totaled.

The use of mathematical sentences in the sigma formula must also be the same as the examples above. where in the formula section is written ui so that in the fractional formula the formula remains the same unchanged.

The following is an example of changing the notation in the sigm formula:

i=312i= i=38i+i=912i

i=312i= 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 75

i=38i= 3 + 4 + 5 + 6 + 7 + 8 = 33

i=912i= 9 + 10 + 11 + 12 = 42

So when you add up i=38i + i=912i the result is 75.

So in changing this index you have to pay attention to the upper limit and the lower limit must have sequential values. In addition, the sentence on the sigma formula fraction must be the same in the use of the symbol.

  • Separation of First and Last Tribes

It’s different with the second way, changing the form of the formula is done by separating the terms in the sigma notation formula. To find out, the following is an example of the formula used.

i=1nui= i=1n-1u1Un

We can try the above formula on the following examples:

i=1n-13i+9= i=1n-13i + 9 + (3n + 9)

So the solution is as follows:

i=3n-13i + 9= 3 + 9 + 6 + 9+9 + 9+…+3n-1+ 9+(3n + 9)

Then we take the last term, namely (3n+9) then the result will be:

3 + 9 + 6 + 9+9 + 9+ + 3n-1+ 9= i=3n-13i + 9 , after that we put the last term back into the addition form.

i=3n-13i+9 +3n+9= i=3n3i+9

Nature Sigma Notation

In order to get to know the properties of addition notation, you can pay attention to some examples of questions below.

  1. k=515003
  2. k=182k
  3. k=210(k2+ 2k)
  4. k=33(k3+6)
  5. k=412(k+3)
  6. k=1001 1006(3k+2)

Here is the solution:

The first property of the sigma formula is k=mn c = (n – m + 1)

k=515003= 3 + 3 + 3 + 3 + + 3

k=51500 3 =(1500-5+1)

k=51500 3 = 1496.3

k=51500 3 = 4488

Problem number two can be solved using the formula for the second property, namely k=mncak=c × k=1nak. So the solution is as follows:

k=182k= 2 k=18k

k=18 2k=2  (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)

k=18 2k =2  36

k=18 2k= 72

To work on problem number three, you can solve it by using the third property of the sigma formula. The following is the formula for the equation k=1n (ak+ bk) = k=mnbk.

k=210k2 +  2k=( k=210k2 + k=2102k)

k=210k2 +  2k= 22+ 32+ 42+ 52+ 2k=210k

k=210k2 +  2k= 4 + 9 + 16 + 25 + 3 ×(2  +  4  +  5 + 6 + 7 + 8 + 9 + 10)

k=210k2 +  2k= 53 + 3 (51)

k=210k2 +  2k= 206

Whereas in question number 4 you can use the properties of the sigma formula k=nn ak = 0. So the result of k=33 (k3 + 6) =0

In the fifth problem, you can use the properties of the sigma formula k=m nak= k=mp-1ak + k=10nak. Then the solution is as follows.

k=412k + 3= k = 46k + 3 + k = 710k + 3

You can solve the sixth problem by using the formula k=mnak = k=m=pn-pak+p . Here is the solution to this problem.

k = 10011006(3k + 2)=  k=1001-10001006-1000[5k +1000 +2]

k = 10011006(3k + 2)=k=16(3k+ 3002)

Examples of Sigma Notation Questions and Answers

To get better at mastering the sigma notation material, of course you need to practice and know more about the sigma formula. Here are some examples of problems that you can look at along with the solution formula.

Question 1

If the value k=121k=x, what is the value of k=100310213k-1995?. To solve it you can use the formula for the third nature.

k= 100310213k – 2999= k=1003-10001021-1000(3(k +1000)-2999)

k= 100310213k – 2999=  3213k+3000-2999

k= 100310213k – 2999=  3213k + 1

Then you can apply the first and second properties as follows.

k = 100310213k – 2999=  k = 3213k  +3211

k = 100310213k – 2999=  3 k = 321k+20-3+3 . 1

k = 100310213k – 2999=   3k + 20

2nd Question

Value for i=150(2i + i2)=X while m=150(2m + m2)=Y. What is the concept of the relationship between M and N? The solution is as follows.

Based on the nature of the sigma formula:

k = mnak= j = mnaj

In this example problem, the two formulas have the same lower bound, which is 1 with a value of n = 50. Even though they have different variables in their functional part, they both have the same form. So it can be concluded that M is equal to N.

This sigma notation material can be developed into various other formulas. So it is very important for you to learn the basic concepts first. In order to be able to apply it when dealing with more complicated formulas.

To master the sigma notation formula better, you can practice consistently. By working on various practice questions and understanding the concept well in determining the formula for solving it.

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Originally posted 2022-12-08 11:44:27.

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