You must have visited a theatre to view a film with friends or family. Have you ever noticed how the seating is often set up at the theatre when purchasing your tickets? There will always be fewer seats in the preceding row than in the one after it by a certain amount. Typically, this seating arrangement follows an arithmetic progression. An arithmetic series is said to decrease or increase by a fixed amount as a result. On the other hand, when we talk about a geometric sequence, it is a totally other entity. Most of you played with a variety of balls when you were kids.
You'll note that whether you use a basketball or a football, the height at which it bounces tends to drop each time it strikes the ground. This decline in bounce height follows a geometric progression. Since each term multiplies or divides by the same value from one specific term to the next, you may claim that the geometric sequence is essentially a series. The common ratio is the quantity by which a phrase divides or multiplies.
Geometric vs. Arithmetic
The key distinction between arithmetic and geometric sequence is that it maintains a constant ratio between its two successive terms. In contrast, an arithmetic sequence maintains a constant difference between its two consecutive terms. The term "common difference" refers to the difference between two successive terms in an arithmetic sequence. On the other hand, the common ratio is the proportion between any two consecutive words in a geometric series.
Mathematical operations that deal with numerical systems and related activities are referred to as arithmetic. It is employed to obtain a single, precise value. The Greek term "arithmos," which means "numbers," is where the English word "arithmetic" originates. The main topics in this branch of mathematics are the study of numbers and the characteristics of frequent operations like addition, subtraction, multiplication, and division.
A sequence is a group of things arranged in a particular order (typically numbers). The two most common kinds of mathematical sequences are arithmetic and geometric sequences. Contrarily, a geometric series has a set ratio between each pair of succeeding terms.
The concept of a series in mathematics is strongly related to sequences. An ordered collection of numbers is a sequence, which might be endless or finite. The difference between any two numbers in an arithmetic progression is always the same. A geometric progression is a series having a fixed ratio of two succeeding numbers. The number of words in these progressions can be countable or uncountable depending on whether they are finite or infinite.
Generally, a series can be described as the collection of elements in a sequence. An arithmetic series is the sum of an arithmetic progression. A geometric series is what is known as the sum of a geometric progression.
Arithmetic and Geometric Mathematics: A Brief History
There is proof that even the earliest members of a man employed arithmetic. The roughly 22,000-year-old Ishango bone from central Africa proves that prehistoric persons had a basic understanding of addition and subtraction. Later, arithmetic systems were established by the Babylonians, Egyptians, Greeks, and Indians. The Indian system, which eventually adopted Arabic numerals, was successful because it takes place value and the concept of zero into account.
The Indus Valley, Egypt, and Mesopotamia are three societies that needed to construct many large-scale structures. These societies provide geometric "proof of the early usage of geometric notions. All of these early geometrical shapes were formalized by Euclid in nine different volumes about 3,000 years later. The geometric legacy was preserved by Arabs during the Middle Ages and was brought back to Europe during the Renaissance. It was developed and helped to bring about calculus.
Arithmetic and Geometry Uses
All other math is built on arithmetic. Essentially, it involves adding, taking away, multiplying, and dividing. It also alludes to the more advanced idea of number theory, which deals with the investigation of integer properties. In addition to being utilized in geometry, algebra, and calculus, arithmetic also serves various practical purposes like checking a checkbook, calculating the tip at a restaurant, budgeting, extending recipes, and much more.
Principles governing shapes and lines are governed by geometry. Therefore, it plays a significant role in construction and architecture. In addition, it is employed in surveying and navigation. Calculus was created using geometrical ideas, as was previously mentioned.
Difference Between Arithmetic and Geometric Sequence in Tabular Form
Parameter of Comparison
It is a list of numbers in which every new term alters from another preceding term by a definite quantity.
It is a sequence of numbers in which each new term is calculated by multiplying by a non-zero and fixed number.
Addition or Subtraction
Multiplication or Division
A constant difference between 2 successive terms.
A common ratio between 2 successive terms.
What is an Arithmetic Sequence?
A list of integers with a distinct pattern is known as an arithmetic sequence. It is an arithmetic sequence if you can subtract any number in the sequence from the one before it and get a value that is either constant or always the same.
The common difference, represented by the letter d, is the constant difference between every pair of consecutive or succeeding numbers in a sequence.
To transition from one phrase to another, we use the common distinction. How? To get to the following term, start with the current term, add the shared difference, and so on. The terms in the sequence are created in this manner.
- The sequence rises if there is a positive difference between two phrases that follow one another.
- On the other hand, we refer to a series as diminishing when the difference is negative.
What is Geometry?
A unique kind of sequence is a geometric sequence. Every term in the series (aside from the first term) is multiplied by a fixed amount to determine the following term. In other words, we multiply the current phrase in the geometric sequence by a constant term (called the common ratio) and then divide the current term in the geometric sequence by the same common ratio to discover the previous term in the geometric sequence. For example, here is an illustration of a geometric sequence with a common ratio of 2, which is 3, 6, 12, 24, and so on. A geometric sequence's common ratio can be either negative or positive, but it cannot be zero. Here, we discover the formulas for the following geometric sequences:
- A geometric sequence's nth term
- The geometric sequence's recursive equation
- a finite geometric sequence's sum
- the total of an endless geometric series
There are infinite and finite geometric sequences. We will study more about the aforementioned geometric sequence formulas in this section, along with examples and proofs.
A unique geometric sequence has a constant ratio between every two succeeding terms. In other words, each phrase in a geometric series is multiplied by a constant to produce the following term. Therefore, a geometric series has the formula a, ar, ar2, where an is the first term, and r is the sequence's common ratio.
How did Geometry Evolve?
Geometry has existed since the time of the earliest men. Although this topic did not exist at the time, the usage of geometrical notions can be seen in the artifacts, ruins, and fossils. For example, the principle of a round item decreasing friction was recently applied when the wheel was created. This is one of the top 5 practical applications of geometry. Even today, we still find driving a car on a round tire very convenient. This is the history of geometry's development and how it came to be considered a subject in ancient Greece.
The Greek civilization saw the greatest expansion of mathematics' geometrical branch. Famous mathematicians and philosophers like Pythagoras, Archimedes, Thales, and Euclid explained the various facets of geometry and laid the groundwork for further developments. The foundation for the ideas we study, which relate to how geometry is used in everyday life, has been built over time by these civilizations.
Thales established numerous mathematical relations and relationships, as well as the fundamentals of geometry. For example, the fact that the sum of a triangle's angles is always 180 degrees was established by Pythagoras. He bears the name of the theorem that describes the connection between a right-angled triangle's base, hypotenuse, and perpendicular.
When he authored volumes on many ideas, Euclid gave geometry a solid foundation in the third century BCE. In his book "The Elements of Geometry," he set the groundwork for numerous parts of geometry that are still in use today. Likewise, his ideas, such as the character of all right angles and the fact that two points can be united to form a straight line, are still in use today.
Benefits of Geometry for Students in Daily Life
Since geometry has many practical applications, it is essential that students learn it. Here are a few examples that illustrate its importance:
- The study of geometry equips students with various fundamental abilities and aids in developing their capacity for logical thought, deductive reasoning, analytical reasoning, and problem-solving. Assisting in their overall development in the process.
- The notion of geometry enables students to relate how classroom objects are mapped to contexts in the real world regarding their location and direction, helping them develop practical thinking.
- Additionally, problem-solving and higher-order thinking skills (HOTS), which geometry enables kids to develop, depend on a grasp of spatial relationships.
- It has numerous real-world applications because it aids in our decision-making about using resources, creating designs, and even the actual construction process itself. As a result, it benefits the kids.
Applications of Geometry in Daily Life
Building Construction, First
Creating houses, dams, rivers, highways, temples, and other structures is the best application of geometry in daily life. Geometry has been skillfully applied for eons to create the temples that house our nation's history. Some well-known temples are even regarded as magical inventions created by people using simple tools.
Graphics on Computers
Geometry is utilized in audiovisual presentations in various contexts, including teaching and entertainment. Computer graphics is one of the most prevalent applications of geometry in daily life that we may take into consideration. Geometrical principles are used in the design of devices like smartphones and laptops. Geometry is also used in the games we play to determine relationships between distance.
The use of geometry by painters to create the best paintings and convey their beliefs is another excellent application of the subject. Using paintbrushes, varied strokes, and colors results in beautiful artwork. Additionally, artists create the clothing, accessories, and other things we use. You can see how crucial geometry is to daily life.
Measuring Planetary Motions and Orbits
Astronomers utilize this idea to follow stars, calculate satellite and planet distances, and measure the distance between them. Additionally, scientists take measurements and plan the launch paths for satellites. One of the best uses of coordinate geometry in daily life is this.
Main Differences Between Arithmetic and Geometric Sequence in Points
- Arithmetic sequences are numbers calculated by taking a fixed term away from or adding it to the previous term. On the other hand, a geometric sequence consists of a series of integers where each new number is produced by multiplying the preceding one by a fixed, non-zero value.
- The common difference, denoted by "d," is the difference between two consecutive terms in an arithmetic sequence. The common ratio, denoted by "r," is the quantity by which terms in a geometric sequence multiply or divide.
- The variation is linear when it comes to an arithmetic series. The variation, however, takes the shape of an exponential in a geometric series.
- Depending on the common difference, the numbers in an arithmetic sequence may move forward positively or negatively. A geometric sequence, however, is exempt from this restriction because the numbers may advance in the sequence alternately in both positive and negative directions.
- Infinite geometric sequences can either converge or diverge, while infinite arithmetic sequences diverge.
The two types of sequences differ substantially from one another, as should be clear from the above description. Arithmetic sequences can also calculate sales, costs, final increments, etc. However, the practical use of geometric sequence is to measure things like interest and population increase.