Introduction to Arithmetic Progression

History of Mathematics: Arithmetic Progression

Arithmetic Progression (AP) is a fundamental concept in mathematics, with applications that date back to ancient civilizations and continue to be useful today. It appears in several areas of knowledge, such as economics, engineering and social sciences. This concept is linked to the idea of ​​sequences of numbers that follow a constant addition rule, called ratio.

Antique

The origin of Arithmetic Progression can be traced back to ancient civilizations, such as the Egyptians and the Babylonians. These cultures already demonstrated knowledge of numerical sequences and sums of simple arithmetic series in contexts such as counting food, harvest records and even astronomical calculations.

Babylonians: The Babylonians, for example, used rudimentary arithmetic techniques on clay tablets dating from approximately 2000 BC. Although their calculation methods were not formalized as in modern mathematics, they had notions of addition and subtraction that are close to the concept of AP, especially when dealing with the growth of resources at the over time.

Greeks: In Ancient Greece, mathematicians such as Euclid (around 300 BC) were responsible for giving more solid foundations to the studies of arithmetic. In his famous work, The Elements, Euclid explored properties of numbers and progressions, establishing principles that would be essential for the construction of the theory of arithmetic progressions.

Middle Ages

During the Middle Ages, Arabic mathematics was crucial to the development of arithmetic. Al-Khwarizmi (780-850 AD), known as the “father of algebra”, introduced a systematization of calculation that also included the understanding of numerical sequences. Although Al-Khwarizmi’s focus was more on algebra, his works prepared the ground for the development of formulas that would later be used in the study of arithmetic progressions.

Renaissance

In the period of the Renaissance, European mathematics experienced a significant renaissance with figures such as Niccolò Tartaglia and Galileo Galilei. These mathematicians began to explore arithmetic progressions with greater scientific rigor. A famous example is Galileo’s use of progressions in describing uniformly accelerated motion. He showed that the distances covered by a body in free fall, in equal time intervals, form an arithmetic progression.

By the 17th century, mathematicians such as Johann Faulhaber and Pierre de Fermat were studying sums of powers and numerical series, contributing to the formal development of sum formulas for arithmetic progressions.

Formalization: 17th and 18th century

The general formulas for arithmetic progressions began to be systematized with the advancement of algebra. The formula for the sum of a PA, for example, is attributed to mathematicians who worked with the development of series.

A curious and famous story is that of Carl Friedrich Gauss (1777-1855), considered one of the greatest mathematicians of all time. It is said that, at the age of 7, Gauss surprised his teacher by quickly adding the numbers from 1 to 100. Instead of adding them manually, he realized he could group the numbers together to simplify the calculation.

The sum $1 + 100 = 101$, $2 + 99 = 101$, $3 + 98 = 101$, and so on.

With this, he discovered the formula for the sum of a PA:

$$S_n = \frac{n(a_1 + a_n)}{2}$$

where $S_n$ is the sum of $n$ terms of the progression, $a_1$ is the first term, and $a_n$ is the last term.

19th and 20th century: Generalizations

During the 19th and 20th centuries, arithmetic progressions were widespread in several branches of mathematics, including number theory and mathematical analysis. Mathematicians such as Joseph-Louis Lagrange and Leonhard Euler also made contributions to the understanding of the sums of series, a concept directly related to progressions.

Arithmetic Progression (AP)

Definition: An Arithmetic Progression (AP) is a numerical sequence in which the difference between consecutive terms is constant. This difference is called PA ratio and is represented by the letter $r$. In formal terms, a sequence of numbers $ (a_1, a_2, a_3, \dots, a_n) $ is an arithmetic progression if, for any term $a_n$, we have:

where:

• $a_1$ is the first term of the progression;
• $r$ is the ratio, that is, the constant difference between consecutive terms.

Example 1: Simple PA

Consider the sequence $2, 5, 8, 11, 14, \dots$. This is a PA with:

• $a_1 = 2$ (first term);
• $r = 3$ (reason, because $5 - 2 = 3$).

Each term is obtained by adding 3 to the previous term.

General Term of a PA

The general term of a PA, that is, the $n$th term, can be obtained by the formula:

where:

• $a_n$ is the $n$-th term of the PA;
• $a_1$ is the first term;
• $r$ is the ratio;
• $n$ is the position of the desired term.

Example 2: Calculation of the $10^\circ$ term

Let the PA be $4, 7, 10, 13, \dots$, with $a_1 = 4$ and $r = 3$. To calculate the $10^\circ$ term, we use the general term formula:

Therefore, the $10^\circ$ term of this PA is $31$.

Sum of Terms of a PA

The sum of the $n$ first terms of an arithmetic progression is given by the formula:

where:

• $S_n$ is the sum of the $n$ first terms;
• $a_1$ is the first term;
• $a_n$ is the $n$-th term.

This formula is derived from the fact that, when adding the terms of a PA, we can group them so that each pair of added terms (first with the last, second with the penultimate, etc.) results in the same sum.

Example 3: Sum of the first 20 terms

Let the PA be $5, 9, 13, 17, \dots$, with $a_1 = 5$ and $r = 4$. To calculate the sum of the 20 first terms, we first determine the $20^\circ$ term:

Now, we apply the formula for the sum of terms:

Therefore, the sum of the first 20 terms of this PA is $860$.

Classification of Arithmetic Progressions

Arithmetic progressions can be classified according to the value of the ratio $r$:

1. Increasing PA: When $r > 0$, that is, the ratio is positive and the terms increase throughout the sequence.

   • Example: $3, 7, 11, 15, \dots$, with $r = 4$.

2. Decreasing PA: When $r < 0$, that is, the ratio is negative and the terms decrease throughout the sequence.

   • Example: $10, 6, 2, -2, \dots$, with $r = -4$.

3. PA Constant: When $r = 0$, that is, all terms in the sequence are equal.

   • Example: $5, 5, 5, 5, \dots$, with $r = 0$.

Applications of Arithmetic Progression

Arithmetic progressions have several practical applications, such as:

• Economics: Calculation of amortization and simple interest, where values ​​grow at regular intervals.
• Physics: Description of uniform movements, such as the free fall of bodies.
• Engineering: Planning production or linear distribution of resources over time.
• Financial Mathematics: Installments and fixed payment plans are often modeled as arithmetic progressions.

Example 4: Financial Application

Suppose that a loan of 1000,00 reais must be paid in 5 monthly installments, with a fixed increase of 50,00 reais in each installment. The first installment is 200,00 reais. The plot values ​​form a PA with $a_1 = 200$ and $r = 50$. The last installment ($a_5$) will be:

The total sum paid over the 5 months is:

Therefore, the total paid at the end of 5 months will be R$ 1500,00.

Conclusion

Arithmetic progression is a fundamental concept that has wide applicability in different areas of knowledge. Its simplicity and practical utility make it an essential tool for both theoretical calculations and everyday problems. The formulas for calculating the general term and the sum of terms are valuable tools for dealing with situations involving linear growth or decrease.