Arithmetic Sequences – In mathematics, there are terms of sequences and series that can be encountered when studying arithmetic material. These sequences and series cannot be separated because they are related to one another. Simply put, an arithmetic sequence is a number with a fixed pattern based on the operations of addition and subtraction. Meanwhile, an arithmetic series is the sum of the first n terms of an arithmetic sequence.
To find out more about arithmetic sequences and series, check out the information in the articles below.
Definition of Arithmetic
Arithmetic table for children, Lausanne, 1835.
Arithmetic (sometimes misspelled as arithmetic, derived from the Greek αριθμός – arithmos = number) or previously called arithmetic is a branch (or precursor) of mathematics that studies the basic operations on numbers. By ordinary people, the word “arithmetic” is often considered a synonym for number theory. Please see numbers for a deeper understanding of number theory.
The prehistory of arithmetic is limited to a small number of artifacts, which can demonstrate the concepts of addition and subtraction, the most famous of which is the Ishango bone from Central Africa, dating to somewhere between 20,000 and 18,000 BC, although its interpretation is disputed.
The earliest written records show the Egyptians and Babylonians using all the basic arithmetic operations as early as 2000 BC. These artifacts do not necessarily reveal the specific process used to solve the problem, but the characteristics of a particular number system greatly influence the complexity of the method. The hieroglyphic system for Egyptian numerals, as well as later Roman numerals, is derived from the counting signs used for counting.
In both cases, this origin returns a value that uses a decimal base, but doesn’t include positional notation. Complex calculations with Roman numerals require the assistance of a counting pad (or Roman swipoa) to produce results.
Early number systems that included non-decimal positional notation, included the sexagesimal (base 60) system for Babylonian numerals, and the vigesimal (base 20) system which determined Mayan numerals. Because of this place value concept, the ability to reuse the same number for different values contributes to a simpler and more efficient method of calculating.
The continuing historical development of modern arithmetic begins with the Hellenistic civilization of ancient Greece, although it dates back much later to Babylonian and Egyptian examples. Prior to the work of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarizes the viewpoints of earlier Pythagorean approaches to numbers, and their relationship to one another, in Introduction to Arithmetic .
Greek numerals were used by Archimedes, Diophantus, and others in a positional notation that does not differ much from modern notation. The ancient Greeks didn’t have a zero symbol until the Hellenistic period, and they used three separate sets of symbols as digits: one set for the ones place, one for the tens place, and one for the hundreds.
For the thousands place, they will reuse the symbol for the ones place, and so on. Their addition algorithms are identical to modern methods, and their multiplication algorithms differ only slightly. The long division algorithm is the same, and the digit-by-digit square root algorithm, popularly used as recently as the 20th century, was known by Archimedes (who may have discovered it).
He preferred it to Heron’s Method of successive approximation because, once calculated, a digit does not change, and the square root of a perfect square, such as 7485692. For numbers with fractional parts, such as 546.934, they use a negative power of 60 instead of a negative power of 10 for fractional part 0.934.
The ancient Chinese had an advanced 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 positional notation similar to that of the Greeks. Because they also lack symbols for zeros, they have one set of symbols for the ones place, and a second set for tens.
For the hundreds place, they then reuse the symbol for the ones place, and so on. Their symbols are based on ancient counting sticks. The exact time when the Chinese began counting by positional representation is unknown, although it is known that adoption began before 400 BC. The ancient Chinese were the first to discover, understand and meaningfully apply negative numbers. This is explained in the Nine Chapters on Mathematical Arts ( Jiuzhang Suanshu ), written by Liu Hui dating from the 2nd century BC.
The gradual development of the Hindu-Arabic numeral system independently created the concepts of place value and positional notation, which incorporated simple methods for computing with a decimal base, and the use of a digit representing 0. This allowed the system to consistently represent large and small integers, a approach that eventually supersedes all other systems.
In the early 6th century AD, the Indian mathematician Aryabhata included existing versions of this system in his work, and experimented with different notations. In the 7th century, Brahmagupta established the use of 0 as a separate number, and determined the results of multiplying, dividing, adding and subtracting zero and all other numbers — except for the product of division by zero. His colleague, the Syrian bishop Severus Sebokht (650 AD) said, “The Indians have a method of calculation that cannot be praised by a single word. Their rational mathematical system, or their method of calculation. I mean the system uses nine symbols”. The Arabs also studied this new method and called it hesab .
Leibniz’s Stepped Reckoner was the first calculator that could perform all four arithmetic operations.
Although the Codex Vigilanus describes an early form of Arabic numerals (omitting the 0) in AD 976, Leonardo of Pisa (Fibonacci) is primarily responsible for spreading their use throughout Europe after the publication of his book Liber Abaci in 1202. He writes, “The method of the Indians ( Latin Modus Indoram ) surpasses any known computational method. That’s an amazing method. They do their computations using nine numbers and a zero symbol.
In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities. The flourishing of algebra in the medieval Islamic world, and also in Renaissance Europe, was the result of simplifying computations via decimals.
Various types of tools have been invented and are widely used to assist in numerical computations. Before the Renaissance, they were a variety of abaci. More recent examples include the shift rule s, nomograms and mechanical calculators, such as Pascal’s calculator. Today, they have been replaced by calculators and electronic computers.
Operations in Arithmetic
The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also covers more advanced operations, such as manipulation of percentages, square root s, exponents, logarithmic functions, and even trigonometric functions, in the same vein as logarithms (prosthaphaeresis). . Arithmetic expressions must be evaluated according to the order in which the operations are intended.
There are several methods of specifying this, either the most common, along with the infix notation, explicitly using parentheses and depending on the order of operation of the priority rule, or using the prefix or postfix notation, which uniquely fixes the order of execution itself. Any collection of objects in which all four arithmetic operations (except division by zero) can be performed, and in which these four operations obey ordinary laws (including distribution), is called a field.
1. Basic Arithmetic Theorem
The fundamental theorem of arithmetic states that any integer greater than 1 has a unique prime factorization (representation of a number as the product of prime factors), excluding the order of factors. For example, the number 252 has only one prime factorization, namely:
- 252 = 2 2 × 3 2 × 7 1
Elements of Euclid himself first introduced this theorem and provided a partial proof (the so-called Euclidean lemma). The basic theorem of arithmetic was first proved by Carl Friedrich Gauss.
2. Number Theory
Until the 19th century, number theory was a synonym for “arithmetic”. The problems it tackles are directly related to elementary operations and those related to primality, division, and the solution of equations in integers, such as Fermat’s last theorem.
It seems that most of these problems, although very basic to state, are very difficult and probably cannot be solved without very deep mathematics involving concepts and methods from many other branches. This led to new branches of number theory such as analytic number theory, algebraic number theory, diophantine geometry, and arithmetic algebraic geometry.
Wiles’ proof of Fermat’s Last Theorem is a typical example of the need for sophisticated methods, far beyond those of classical arithmetic, to solve problems that can be expressed in elementary arithmetic.
Understanding Arithmetic Sequences and Series
In mathematics, arithmetic sequences and series, also known as arithmetic sequences and series, are sequences that have a certain pattern, that is, the difference between two successive terms is the same and remains the same. In other words, each term (except the first term) in an arithmetic sequence is obtained from the previous term by adding a fixed number, for example:
This arithmetic sequence can be expressed by the following formula.
Furthermore, as adapted from the book entitled Mathematics SMK 2: Business and Management Group published by Grasindo, an arithmetic sequence is a line where the value of each term is obtained from the previous term by adding or subtracting a number b.
Furthermore, the difference between the values of the terms close to each other and is always the same, namely b. For example:
Un – U(n-1) = b
For example, rows 1, 3, 5, 7, 9 are arithmetic rows with values:
b = (9 – 7) = (7 – 5) = (5 – 3) = (3 – 1) = 2
Meanwhile, an arithmetic series is a sum between terms of an arithmetic sequence. For the sum of the first terms to the nth term, the arithmetic sequence can be calculated as:
Sn = U1 + U2 + U3 + …. +U(n-1)
or
Sn = a + (a + b) + (a + 2b) + …. + (a + (n – 2)b) + (a + (n – 1)b)
If only the value of a is known, the first term and its value are the nth term, then the value of the arithmetic series is:
Sn = n/2(a + Un)
Arithmetic Sequence
-
Proof We start sorting it from tribe . We continue for the 2nd, 3rd, to .
1. Different
- .
2. Middle Tribe
with
- .
We can expand it again to get:
- .
Arithmetic Sequence and Series Formulas
After discussing the brief understanding of arithmetic sequences and series, understand the following description of the formulas, quoted from a book entitled Complete Collection of Mathematics Formulas for SMA/MA IPA/IPS by Khoe Yao Tung, along with the information.
1. Arithmetic Sequence Formulas
The formula for determining the nth term of an arithmetic sequence:
Un = a + (n – 1)b or Un = Un-1 + b
Apart from finding the formula for the nth term, the formula used to find the middle value of an arithmetic sequence is:
Ut = ½ (a + Un)
Information:
Un = the nth term
a = U1
Un-1 = the term before the nth term
b = different
2. Arithmetic Series Formulas
At first glance, an arithmetic series has the same formula components as an arithmetic sequence. The difference is that the arithmetic series formula is used to find the desired term, while the arithmetic series looks for the sum of these terms.
For more details, here is the arithmetic series formula, namely:
Sn = n/2 (a + Un) = n/2(2a + (n – 1)b)
Based on this formula, the nth term can be found in the following way, namely:
Un = Sn – Sn-1
Information:
Un = the nth term
a = U1
Un-1 = the term before the nth term
b = different
Examples of Arithmetic Sequences and Series Problems
In order to understand more clearly about arithmetic sequences and series, first look at the example questions below, as quoted from a book entitled Mathematics Isolation for Middle School Grades 1, 2 and 3 by Herlik Wibowo.
Problem 1
An arithmetic series is 5, 15, 25, 35, …. What is the sum of the first 10 terms of the arithmetic sequence?
Given:
n = 10
U1 = a = 5
b = 15 – 5 = 25 – 15 = 10
Answer:
Sn = (2a + (n-1) b )
S10 = ( 2.5 + (10 -1) 10)
= 5 ( 10 + 9.10)
= 5 x 100 = 500
So, the sum of S10 in the arithmetic series, i.e. 500.
Problem 2
Given an arithmetic series with the first term being 10 and the sixth term being 20. Then, determine:
The difference is the arithmetic series.
Write down the arithmetic series.
The sum of the first six terms of the arithmetic sequence.
Answer:
The difference in the arithmetic series is:
Un = a+(n-1)b
U6= a+(6-1) b
20= 10+(5)b
b= 10/5 = 2
So, the difference in the arithmetic series is 2.
The arithmetic series, namely:
10+12+14+16+18+20+…+Un
The sum of the sixth term, S6 is:
Sn =n/2 (2a+(n-1) b)
S6= 6/2 (2.10+(6-1) 2)
=3(20+10)
=90
So, the sum of the sixth term the series is 90.
Problem 3
The 40th term of sequence 7, 5, 3, 1, … is…
Given:
a = 7
b = -2
Answer:
Un = a + (n – 1)b
U40 = 7 + (40-1)(-2)
= 7 + 39 . (-2)
= 7 + (-78)
= – 71
So, the 40th term of the arithmetic sequence is –71.
Problem 4
In a theater, the seats are arranged with the front row containing 12 seats, the second row containing 14 seats, the third row containing 16 seats, and so on. The number of seats in the 20th row is…
Given:
a = 12
b = 2
Answer:
Un = a + (n – 1)b
U20 = 12 + (20-1)2
= 12 + (9)2
= 12 + 38
= 50
So, the number of seats in the 20th row is 50 seats.
Problem 5
A petty employee receives a first year’s salary of Rp. 3,000,000.00. Every year the salary increases by IDR 500,000.00. The amount of money received by the employee for ten years is…
It is known:
First salary = a = IDR 3,000,000.00
Salary increase every year = b = IDR 500,000
Tenth year salary = U10
Total salary for ten years = S10
Answer:
Un = a + (n – 1)b
U10 = 3,000,000 + (10 – 1)500,000
= 3,000,000 + (9 × 500,000)
= 3,000,000 + 4,500,000
= 7,500,000
So, employee salaries obtained in the tenth year is IDR 7,500,000.00
Problem 6
Calculate the sum of the values for the 4th term (S4) of the arithmetic series if there are numbers: 4, 8, 16, …?
Given:
a = 4
b = 8-4 = 4
n = 4
Answer:
Un = a + (n-1) b
Un = 4 + (4-1)4
Un = 4 + 12
Un = 16
Then, how much Sn?
Sn = 1/2 n ( a + Un )
S4 = 1/2 .4 (4 +16)
S4 = 4/2 (20)
S4 = 40
So, the sum of the values of the 5th term in an arithmetic sequence is 40.