Arithmetic Series – Students who have entered grade 9 at the Junior High School (SMP) level of education must prepare themselves to take the National Examination (UN). For this, they can no longer play games in learning and must increase their focus on education.
There are at least 4 subjects that students must study for this UN. These subjects are Indonesian, English, Mathematics, and Natural Sciences (IPA). Even though the level of difficulty of the 4 subjects is the same, it cannot be denied that one of the biggest scourges for students is mathematics.
Besides they have to repeat the lessons they got in grades 7 and grade 8, they will also receive a number of new materials. The level of difficulty of these materials is also quite complicated. One of them is an arithmetic series.
In this article, Sinaumed’s will get an explanation about arithmetic series. Hopefully, this article can be useful for Sinaumed’s who need knowledge related to the topic of arithmetic series, regardless of your background.
Arithmetic progression
Before we enter into the discussion of formulas regarding arithmetic series, it helps us to first understand the definition of a mathematical sequence. Although they are different topics, they have continuity with one another, so students should study both of them in order to have a deep understanding.
Basically, an arithmetic sequence is a sequence of numbers such that the difference between successive terms remains constant. While an arithmetic series can be described as the sum total of the terms in the arithmetic sequence.
The explanation above may not necessarily be understood by some people. Therefore, we will now look at some examples of arithmetic sequences together to make it easier for you to understand this topic.
4,7,10,13,16,19,22…..
The example above is an arithmetic sequence. If Sinaumed’s pays attention, each number in the arithmetic sequence is the number obtained after adding the number 3. Let’s look at other examples of arithmetic sequences.
5,9,13,17,21,25,29…..
This is another example of an arithmetic sequence. Sinaumed’s can see that every number you find in the arithmetic sequence is obtained after adding the number 4. You can also find other examples of arithmetic sequences with the same principle.
After understanding the examples of arithmetic sequences, Sinaumed’s should be able to get a more detailed picture of arithmetic series. We’ll take some of the examples above and turn them into an arithmetic series. Here’s an example:
S 91 = 4,7,10,13,16,19,22= 91
The example above is a simple example of an arithmetic series. So, all Sinaumed’s needs to do is add up the entire arithmetic sequence until you can get a result. Let’s use the above example of an arithmetic sequence as an arithmetic series.
S 119 = 5,9,13,17,21,25,29 = 119
Based on these 2 examples, Sinaumed’s could conclude that the arithmetic series works in the same way. It’s just that, there are times when Sinaumed’s needs to find certain numbers or variables. We will discuss this topic together in the session on the arithmetic series formula and examples of this arithmetic series problem.
History of Arithmetic Series
A bit of history regarding the arithmetic series, there is no definite source regarding who was the person who first discovered the arithmetic series system. However, some sources say that it was Carl Friedrich Gauss, a mathematician and natural science expert from Germany who was the first to find out about arithmetic series.
At that time, Carl Friedrich Gauss was still in elementary school (SD). That said, he reinvented this method for calculating the sum of integers from 1 to 100, by multiplying the n/2 pairs of numbers added up by the value of each pair n+1.
Even so, the above facts cannot be proven true, bearing in mind that there are still other mathematicians who are also famous for creating various mathematical theories even before entering the century AD. Call it figures like Pythagoras and Archimedes from Greece, Aryabhata and Brahmagupta from India, and Zhang Qiujian from China.
Mathematics is indeed something that is considered difficult by most students. Even so, there are times when we find some interesting methods to learn mathematics. One way is to read books that are colorful and easy to understand, such as the book “Junior Scientist Encyclopedia: Mathematics”.
Arithmetic Series Formulas
Armed with the understanding that we have discussed above, now Sinaumed’s can learn the formula for an arithmetic series. You need to know that like a number of other mathematical materials, arithmetic series also has its own variables that students need to pay attention to.
These variables are also related to arithmetic sequences, where sometimes Sinaumed’s needs to find these variables before he can work on arithmetic series problems. Some of the variables in question are the difference and the middle term.
So, below, besides we will learn the arithmetic series formula, we will also learn the difference formula and the middle term formula which you can use if needed. Check out the explanation below.
Difference Formula
b = Un – Un -1
- b = Different
- U n = the umpteenth term
- U n-1 = The nth term minus 1
The difference in an arithmetic sequence or arithmetic series is symbolized by “b”. To find the difference, Sinaumed’s only needs to subtract the uth term (U n ) in the arithmetic row with the other term that comes before the umpteenth term (U n-1 ).
Central Tribal Formula
U t = (a + U n ) ÷ 2
- U t = Middle term
- a = initial term
- U n = Final term
The middle term in an arithmetic sequence or arithmetic series is symbolized as “U t “. To find the middle term, Sinaumed’s must first determine the initial term (a) and the final term (U n ) in the arithmetic row, then divide by 2.
Arithmetic Series Formulas
S n = ½n (2a + (n – 1) b)
- S n = Arithmetic series
- a = First term
- n = Number of terms
- b = Different
The arithmetic series in this formula is symbolized as “S n “. The first thing you need to do to be able to find an arithmetic sequence is multiply half by the number of terms (n) that can be found in the arithmetic sequence.
Next, Sinaumed’s can add up the result of the sum above with the other variables found in the brackets. There are several additions that you have to calculate, namely multiplying 2 by the first term (a), then adding the result of the number of terms that have been reduced by 1 and multiplied by the difference (b).
There are several other arithmetic series formulas that Sinaumed’s can use depending on needs. Luckily, these formulas don’t differ much from the arithmetic series formulas above, and they won’t be used very often. You can read these formulas in the book “Super Complete Core Material & Middle/MTS Mathematics Formulas 7,8,9”.
Example Problem Arithmetic Series
Of course, discussing formulas alone will not have any effect on students’ understanding of mathematics. In order for Sinaumed’s to understand and be proficient in working on math problems, of course you have to continue to practice your skills by doing practice questions.
By practicing math problems, you can test the formulas that you previously learned, and determine whether or not the formula is correct. If you only study math formulas without doing questions, you cannot know whether the formula is right or wrong, and whether it can be used or not.
For this reason, this time we will discuss together several problems related to arithmetic series. These questions will be sorted from the easiest to the most difficult questions, so that your understanding can slowly increase.
If Sinaumed’s wants to try increasing the difficulty level and wants to test whether or not you understand arithmetic series questions, you can do these questions first before reading the discussion below. Happy practicing, Sinaumed’s!
First Question
Consider an arithmetic series that has 8 terms. The first term of this arithmetic sequence is 5, and each term has a difference of 4. Find the arithmetic sequence based on this information!
Sinaumed’s, who read the problem carefully, probably already realized that you can find all the variables. The next task is just to enter these variables into the formula that we have discussed above.
S n = ½n (2a + (n – 1) b)
S n = ½ 8 ((2 x 5) + (8 – 1) 4)
S n = 4 (10 + 28)
S n = 142
The answer above is actually enough. However, there are times when you may not be sure of the answers you have found. If that were the case, Sinaumed’s could have tried to write down the contents of the arithmetic series and ensure that the sum of the arithmetic series was correct. Here’s an example:
S 142 = 5,9,13,17,21,25,29,33
Thus, the results above indicate that your calculations are correct. The result of the arithmetic series in the first question is 142.
Second Problem
An arithmetic series with 12 terms if added together has a final result of 306. How different does this arithmetic series have if the first term is 9?
Sinaumed’s needs to understand that you can’t use the formula to find the difference that was taught earlier. This is because there are no other terms in the problem. Even so, Sinaumed’s doesn’t need to worry because what you need to do is the same as the previous problem.
Sinaumed’s only needs to read carefully which variables have been found in the problem. If you have found what the variables in the problem above are, all you have to do is enter them into the formula we studied earlier.
S n = ½n (2a + (n – 1) b)
306 = ½ 12 ((2 x 9) + (12 – 1) b)
306 = 6 (18 + 11b)
306 = 108 + 66b
306 – 108 = 66b
198 = 66b
198 ÷ 66 = b
3 = b
The calculation of the arithmetic series above is indeed quite long. It is possible if you find errors when calculating the questions above. If Sinaumed’s still has time, you can try to determine whether your calculation is correct or not, by making an arithmetic series based on the information above.
S 306 = 9,12,15,18,21,24,27,30,33,36,39,42
If all the numbers above are added up, the result is in accordance with the final result of the arithmetic series mentioned above. With this, you have proven that your calculations are correct. The difference from the second problem arithmetic series is 3.
Third Problem
The arithmetic sequence U 1 ,U 2 ,U 3 ,U 4 ,U 5 …54.58 is known to have a middle term of 30 and a total of 450 in the arithmetic sequence. How many terms are in the arithmetic sequence above?
As promised, the last problem is quite complicated to solve. There is a lot of incomplete information to find out the final result of the problem. However, this can be an opportunity for Sinaumed’s to use all the formulas that you have learned before.
First, you can find the difference from the arithmetic sequence first. This is because apart from the formula being simple and calculating quickly, you also need a difference to calculate other things in the problem. Here’s the solution.
b = U n – U n-1
b = 58 – 54
b = 4
After finding the difference, we’ll move on to the next step, which is finding the first term in the arithmetic sequence. This first term is important for later finding the total terms of the arithmetic series as a whole. Here, because there is information about the middle term, Sinaumed’s can use the formula to find the initial term.
U t = (a + U n ) ÷ 2
30 = (a + 58) ÷ 2
30 = a/2 + 29
30 – 29 = a/2
1 = a/2
1 x 2 = a
2 = a
After you have found all the variables, the last step you need to do is enter all these variables into the arithmetic series formula to find how many terms are in the arithmetic sequence and also this arithmetic series. The calculation is as follows:
S n = ½n (2a + (n – 1) b)
450 = ½n ((2 x 2) + (n – 1) 4)
450 = ½n (4 + 4n – 4)
450 = 4n + 2n² – 4n
450 = 2n²
450 ÷ 2 = n²
225 = n²
15 = n
Calculations of this length can tire Sinaumed’s so that your focus drops in the middle of the calculations. It would be nice if Sinaumed’s double-checked whether or not the results you got from this calculation were correct. The method is quite simple, that is, as you did in the previous questions.
S 450 = 2,6,10,14,18,22,26,30,34,38,42,46,50,54,58
With this proof, you have succeeded in showing that your calculations are correct, because the number of terms in this arithmetic sequence and mathematical series is 15.
The questions above can be a scourge for Sinaumed’s who are not used to working on arithmetic series problems like this. However, if you practice a lot, you will get used to it and in the end you will be able to work on the questions smoothly. The book “Super Coach for Independent Student Learning Patterns for SMP/MTS Class IX Mathematics” besides containing mathematical formulas and explanations, also has questions that can help improve your understanding.
If Sinaumed’s is interested in buying math problem books, or looking for other articles related to the same topic, Sinaumed’s can directly visit the www.sinaumedia.com site . There, you will find reading material from sinaumedia, #Friends Without Limits, which are useful to read in order to gain knowledge, insight, and also #MoreWithReading information.