Example of an arithmetic sequence – Hello Sinaumed’s friends , can you continue the sequence of numbers 1, 3, 5, 7, 9, 11, 13, …, …, …? Come on, what number fills the three gaps? Yes, that is true. The three gaps are filled with the numbers 15, 17 and 19. Easy, right? Are you now trying to figure out which number is the number 20? Well, you must be dizzy? If you want to find numbers in a large enough set, you can use the formula for an arithmetic sequence.
In mathematics there is the term arithmetic sequence which we can encounter when studying arithmetic material. Simply put, an arithmetic sequence is numbers with a fixed pattern based on the operations of addition and subtraction. What is meant by an arithmetic sequence? Come on, see the explanation!
Definition of Arithmetic Sequences
An arithmetic sequence is a sequence of signs or numbers with a fixed difference. For example, as at the beginning of this article, namely the sequence of numbers 1, 3, 5, 7, 9, 11, 13, 15 and so on. If you pay attention, the difference in these numbers is always the same, namely 2. The difference in an arithmetic series is called the difference, or expressed mathematically by b. Each number that makes up the series is called a term, or expressed as an Un. For example 1 = 1st term (U1), 3 = 2nd term (U2), 5 = 3rd term (U3), etc. While the first term of the sequence (U1) is expressed mathematically as a.
Note the following explanation:
The number sequence above has a difference or difference of 3 between two successive sequence terms. So, the sequence of numbers is an arithmetic sequence.
Know the sequence of numbers
This number sequence has a fixed difference or difference between two successive series terms, namely -4. This means that the sequence of numbers is an arithmetic sequence.
From these two descriptions we can conclude that an arithmetic series has a fixed difference (often indicated by b).
If b is positive, the arithmetic series is said to be an increasing arithmetic series. On the other hand, when b is negative, the arithmetic series is said to be a decreasing arithmetic series.
Arithmetic Sequence Formula
You understand ascending and descending arithmetic sequences. How do you find one of the terms in a sequence if you only know the first term and its difference?
How to find the difference when only the first term and one other term are known? To answer this, study the following description.
The arithmetic series is known as follows. U1, U2, U3, U4, U5, U6, …, Un – 1, Un
From the received path
U1 = a (first term is denoted by a)
U2 = U1 + b = a + b
U3 = U2 + b = (a + b) + b = a + 2b
U4 = U3 + b = (a + 2b) + b = a + 3b
U5 = U4 + b = (a + 3b) + b = a + 4b
U6 = U5 + b = (a + 4b) + b = a + 5b
Un = Un − 1 + b = (a + (n − 2) b ) + b = a + (n − 1) b
So the formula for the nth arithmetic sequence can be written as follows.
To find differences in an arithmetic sequence, try to pay attention to the following explanation.
U2 = U1 + b then b = U2 − U1
U3 = U2 + b then b = U3 − U2
U4 = U3 + b then b = U4 − U3
U5 = U4 + b then b = U5 − U4
Un = Un − 1 + b then b = Un − Un − 1
Therefore, the difference of an arithmetic sequence is expressed as follows.
Second Degree Arithmetic Sequence Formulas
To find Un in the unit arithmetic sequence, the formula is the same as the arithmetic row formula previously studied, namely Un = a + (n-1)b. So, to find Un in an arithmetic series with two and three levels, you can use the following formula.
Now let’s try to find a two-level sequence from this formula.
Substitute n = 1 in Un = an2 + bn + c, we get the first term, namely:
Un = an2 + bn + c
U1 = a(1)2 + b(1) + c
U1 = a + b + c
Substituting n = 2 in Un = an2 + bn + c, we get the second term, namely:
Un = an2 + bn + c
U2 = a(2)2 + b(2) + c
U2 = 4a + 2b + c
Substituting n = 3 in Un = an2 + bn + c, we get the third term, namely:
Un = an2 + bn + c
U3 = a(3)2 + b(3) + c
U3 = 9a + 3b + c
Substituting n = 4 in Un = an2 + bn + c, we get the fourth term, namely:
Un = an2 + bn + c
U4 = a(4)2 + b(4) + c
U4 = 16a + 4b + c
This gives the following arithmetic sequence:
If we then look for the difference (difference) of these terms, we get:
The difference between the first term (U1) and the second term (U2).
b = U2 – U1 = (4a + 2b + c) – (a + b + c)
b = 4a – a + 2b – b + c – c
b = 3a + b
The difference between the second term (U2) and the third term (U3).
b = U3 – U2 = (9a + 3b + c) – (4a + 2b + c)
b = 9a – 4a + 3b – 2b + c – c
b = 5a + b
The difference between the third term (U3) and the fourth term (U4).
b = U4 – U3 = (16a + 4b + c) – (9a + 3b + c)
b = 16a – 9a + 4b – 3b + c – c
b = 7a + b
So it is the difference between adjacent terms in an arithmetic sequence
So, since we are looking for a two-order arithmetic sequence using the two-order arithmetic series formula, you can see that the difference of the terms is not constant or the same. So let’s consider 3a + b, 5a + b and 7a + b the new first degree terms. Then we look again for differences between the new terms to find differences that persist at the second level.
The difference between the first term of the first stage (U1*) and the second term of the first stage (U2*).
b = U2* – U1* = 5a + b – (3a + b)
b = 5a – 3a + b – b = 2a
Difference in semester 2 S1 (U2*) with semester 3 S1 (U3*).
b = U3* – U2* = 7a + b – (5a + b)
b = 7a – 5a + b – b = 2a
So the difference between adjacent new terms of degree 1 in arithmetic sequence is:
So now you can see that at the second level we can get a fixed difference of 2a.
So why are we looking for a two-level arithmetic sort model like the one in the image above? The goal is to make it easier to obtain the values a, b and c contained in the two-step arithmetic series formula (Un = an2 + bn + c).
Examples of Arithmetic Problems
Example 1
Mr. Topik opened a catfish pecel stall. On the first day it opened, Topik provided 20 catfish. On the second day, the supply of catfish was increased to 24 catfish. On the third day, the supply was 28. The first week of opening, the number of catfish added with fixed additions. How many catfish did Topic provide on the seventh day?
Discussion:
Is known:
First day supply of catfish (U1) = a = 20
Second day supply of catfish (U2) = 24
Third day supply of catfish (U3) = 28
Asked: U7 =…?
Discussion:
First, you have to find the difference.
b = 24 – 20 = 4 catfish
Therefore,
So, the number of catfish provided by Topik on the seventh day was 44 catfish.
Example 2
The arithmetic sequence is known as follows.
10, 13, 16, 19, 22, 25, ….
Define:
- types of arithmetic sequences,
- the twelfth term of the sequence.
Answer:
To determine the type of arithmetic sequence, determine the value of the difference in the sequence.
b = U2 − U1
= 13 − 10
= 3
Since b > 0, the arithmetic sequence is an ascending arithmetic sequence.
To find the twelfth term (U12), do the following.
Un = a + (n − 1)b then
U12 = 10 + (12 − 1) 3
= 10 + 11 3
= 10 + 33
= 43
So, the twelfth term of the sequence is 43.
Example 3
The first term of the arithmetic sequence is 6 and the seventh term is 24.
- Define line difference.
- Write down the first ten terms of the series. Answer:
Known as:
first term = a = 6
seventh term = U7 = 36
- a) Determination of the difference:
Un = a + (n − 1) b then
U7 = 6 + (7 − 1) b
36 = 6 + 6b
36 − 6 = 6 b
30 = 6b
b = 5
So the difference in the order is 5.
- b) With the first term 6 and a difference of 5 the following arithmetic series is obtained.
6, 11, 16, 21, 26, 31, 36, 41, 46, 51
Given an arithmetic series:
−8, −3, 2, 7, 12, 17, …
Find the formula for the nth term that applies to the series. Answer:
Known as:
a = U1 = −8
b = U2 – U1
= −3 − (−8)
= −3 + 8
= 5
So the general formula of this line is
Un = a + (n − 1) b
= −8 + (n − 1) 5
= −8 + 5n − 5
= 5n − 13
Example 4
Every month, Ziaggi always saves money in the bank. In the first month, he saved Rp. 10,000.00, the second month he saved Rp. 11,000.00, the third month he saved Rp. 12,000.00. And so on, he always saved more than Rp. 1,000.00 every month.
- State the money Ziaggi has saved (in thousands of rupiah) for the first 8 months.
- Determine the amount of money Ucok saved in the 12th month.
Answer :
In thousands of rupiah, the money Ucok has saved for the first 8 months is as follows.
10, 11, 12, 13, 14, 15, 16, 17
It is known that: U1 = 10
b = 1
U12 = a + (n – 1) b
= 10 + (12 – 1) 1
= 10 + 11
= 21
So, the money Ziaggi saved in the 12th month was IDR 21,000.00.
Example 5
In a theater, the seats are arranged so that the first row has 12 seats, the second row has 14 seats, the third row has 16 seats, and so on, always two more. The number of seats in the 20th row is…
Answer:
Example: Un = the number of digits in the nth line
Known as:
U1 = 12,
U2 = 14 and
U3 = 16
Requested:
U20?
Diploma:
The number of seats in each row forms an arithmetic series with a = 12 and b = 2.
So Un = a + (n – 1)b
U20 = 12 + (20 – 1)2
= 12 + (19)2
= 12 + 38
= 50
History of the Discovery of Arithmetic Formulas
Arithmetic (sometimes misspelled as arithmetic, derived from the Greek word ναριμος – arithmos = number), or earlier arithmetic, is the branch (or ancestor) of mathematics that studies the basic operations on numbers. The word “arithmetic” is often viewed by laypeople as synonymous with number theory. Consider numbers for a deeper understanding of number theory.
The prehistory of arithmetic was limited to a small number of objects that could demonstrate the concepts of addition and subtraction. The most famous are the Ishango bones from central Africa, found between 20,000 and 18,000 BC. BC, although the interpretation is disputed. .
The earliest written records show that the Egyptians and Babylonians lived as far back as 2000 BC. using all the basic arithmetic. These artifacts do not necessarily reveal the specific processes used to solve the problem, but the characteristics of a particular number system have an important effect on the complexity of the method. The hieroglyphic system for Egyptian numerals, as well as later Roman numerals, derives from the numbers used in counting.
In both cases, this origin sends a value that uses decimals but doesn’t contain wildcards. Complex calculations involving Roman numerals require the help of a spreadsheet (or Roman swipoa) to get the results.
Early number systems that included non-decimal place notation included the hexadecimal (base 60) system for Babylonian numerals and the vigesimal (base 20) system which determined Mayan numerals. Thanks to this place value concept, the ability to reuse the same number for different values promotes a simpler and more efficient way of calculating.
The continuing historical development of modern arithmetic begins with the Hellenistic civilization of ancient Greece, although it goes back to much later Babylonian and Egyptian examples. Before the work of Euclid around 300 BC. The study of Greek mathematics intersected with philosophical and mystical beliefs. For example, in Introduction to Arithmetic, Nicomachus summarizes the perspectives of Pythagoras’ earlier approaches to numbers and their relationship to one another.
Archimedes, Diophantus and others used Greek numerals for positional notation, which is not much different from modern notation. The ancient Greeks did not have a zero symbol until the Hellenistic period, and used three separate sets of symbols as numbers:
one set for the first digit, one for the tens digit, and one for the hundreds digit.
For thousands of places, they reuse thousands of places symbols, etc. Their addition algorithms are identical to modern methods, and their multiplication algorithms differ only slightly. The long division algorithm is the same, and Archimedes (who may have invented it) knew the number-by-number square root algorithm, which was still widely used in the 20th century.
He preferred Heron’s method of successive approximation because numbers, once calculated, are immutable and are the square root of a perfect square such as 7485692. For numbers with fractions such as 546.934, use the negative power of 60 of the number instead of the negative power of 10 of the fraction 0.934 .
The ancient Chinese had a complex study of arithmetic dating back to the Shang Dynasty and continuing into the Tang Dynasty, from elementary numbers to advanced algebra. The ancient Chinese used place names similar to those of the Greeks. Since they also don’t have symbols for zeros, they have one set of symbols for ones and another sheet for tens.
Then, for hundreds of places, symbols are used again for hundreds of places, and so on. Their symbols are based on the old slide rules. The exact date when the Chinese started counting based on place representation is unknown, although adoption probably occurred before 400 BC. BC begins The ancient Chinese were the first to discover, understand and use negative numbers. This is explained in the Nine Chapters on Mathematical Arts (Jiuzhang Suanshu) taught by Liu Hui in the 2nd chapter. century BC was written.
The gradual development of the Hindu-Arabic numeral system independently created the concepts of place value and place notation, which included simple methods of counting with decimals and the use of a number representing zero. It allowed the system to represent large and small integers constantly, an approach that would eventually supersede all other systems.
As early as the 5th century, the Indian mathematician Aryabhata incorporated existing versions of this system into his work and experimented with various notations. In the 7th century Brahmagupta introduced 0 as a separate number and determined the product of multiplication, division, addition and subtraction of zero and all other numbers – except the product of division by zero. His colleague, the Syrian bishop Severus Sebok Gt (650 AD), said: “The Indians have a method of calculation that cannot be praised in a single word. The existence of a rational mathematical system or method of calculation. I mean, the system uses nine symbols.” The Arabs also studied this new method and called it Hasan.
Although the Codex Vigilanus describes an early form of Arabic numerals (without the 0) in 976, Leonardo de Pisa (Fibonacci) is primarily responsible for spreading their use throughout Europe after publishing his Liber Abaci in 1202. He wrote: The Method of the Indians ( in Latin Modus Indoram) surpasses all known calculation methods. This is an extraordinary method. They perform calculations using nine numbers and a zero symbol.
In the Middle Ages, arithmetic was one of the seven arts taught in universities. The growth of algebra in the Islamic Middle Ages and also in Renaissance Europe was due to the simplification of decimal numbers.
Various tools have been invented and are widely used to assist numerical calculations. Before the Renaissance, they were separate monasteries. More recent examples are transfer rules, nomograms and mechanical calculators such as Pascal’s calculator. Today they have been replaced by pocket calculators and electronic calculators.
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