Diferencia entre secuencia aritmética y geométrica (con tabla)

All of you must have been to movie theaters to watch movies with your friends or family members. While booking your tickets, have you ever noticed the way the seating arrangements are normally made at the movie theater? The number of seats in the previous row will always be lesser than the next row by a specific number.

This seating arrangement is normally in an arithmetic sequence. Thus, it can be said that a sequence that decreases or increases by a constant number is known as an arithmetic sequence. On the other hand, a geometric sequence is something completely different. Most of you have played with some sort of balls during your childhood days.

Whether you use a football or a basketball, you will notice that the height at which it bounces tends to decrease every time it hits the ground. This decrease in the bouncing height is in a geometric sequence.

Thus, it can be said that the geometric sequence is basically a sequence in which each term multiplies or divides by the same value from one specific term to the next one. The value by which a term divides or multiplies is known as the common ratio.

Secuencia aritmética vs geométrica

los difference between Arithmetic and Geometric Sequence is that while an arithmetic sequence has the difference between its two consecutive terms remains constant, a geometric sequence has the ratio between its two consecutive terms remains constant.

The difference between two consecutive terms in an arithmetic sequence is referred to as the common difference. On the other hand, the ratio of two consecutive terms in a geometric sequence is referred to as the common ratio.

Comparison Table Between Arithmetic and Geometric Sequence

Parámetro de comparaciónSecuencia aritméticaSecuencia geométrica
DefiniciónEs una lista de números, en la que cada término nuevo cambia de otro término anterior en una cantidad definida.Es una secuencia de números en la que cada nuevo término se calcula multiplicando por un número fijo y distinto de cero.
Calculado porSuma o restaMultiplicación o división
Identificado porUna diferencia constante entre 2 términos sucesivos.Razón común entre 2 términos sucesivos.
FormarForma linealForma exponencial

¿Qué es la secuencia aritmética?

When you talk about arithmetic sequence or arithmetic progression, it basically refers to a sequence of different numbers in which the difference between 2 consecutive numbers is always constant.

En este tipo de secuencia, diferencia significa el primer término restado del segundo término. Si considera una secuencia como 1, 4, 7, 10, 13 ... es una secuencia aritmética en la que la diferencia constante es 3.

Just like anything else in mathematics, an arithmetic sequence also has a formula. The formula used to find an arithmetic sequence is a, a+d, a+2d, a+3d, and so on. In this formula, “a” is the first term and “d” is the common difference between 2 consecutive terms.

It is important for you to know that the behavior of an arithmetic sequence depends a lot on the common difference. If the common difference or the “d” in the formula is positive, then the terms will grow in a positive manner. However, if the common difference is negative, the terms will grow in a negative manner.

¿Qué es una secuencia geométrica?

The geometric sequence or geometric progression in mathematics happens to be a sequence of different numbers in which each new term after the previous is calculated by simply multiplying the previous term with a common ratio. This common ratio is a fixed and non-zero number. As an example, the sequence 3, 6, 12, 24, and so on is a geometric sequence with the common ratio being 2.

Una secuencia geométrica también tiene una fórmula propia. La forma normal de una secuencia geométrica tiene la forma a, ar, ar², ar³, ar4 y así.

Cuando necesite encontrar el n-ésimo término en cualquier secuencia geométrica, la fórmula a utilizar esnorte = arn-1, where the common ratio “r” and the initial value “a” will be given. There are certain factors you should remember when it comes to a geometric sequence. If the common ratio is positive, the terms will also be positive.

However, if the common ratio is negative, the terms will be alternate between negative and positive. If the common ratio is greater than 1, the growth will be in an exponential form towards positive or even negative infinity. If the common ratio is 1, then the progression will be a constant sequence.

Principales diferencias entre secuencia aritmética y geométrica

  1. Una secuencia aritmética es una secuencia de números que se calcula restando o sumando un término fijo al término anterior. Sin embargo, una secuencia geométrica es una secuencia de números en la que cada nuevo número se calcula multiplicando el número anterior por un número fijo y distinto de cero.
  2. La diferencia entre dos términos consecutivos en una secuencia aritmética se conoce como la diferencia común que está representada por “d”, y el número por el cual los términos se multiplican o dividen en una secuencia geométrica se conoce como la razón común representada por “r”.
  3. Cuando se trata de una secuencia aritmética, la variación es lineal. Por otro lado, cuando se trata de una secuencia geométrica, la variación es exponencial.
  4. En una secuencia aritmética, los números pueden progresar de manera positiva o negativa dependiendo de la diferencia común. Mientras que, en una secuencia geométrica no existe una regla tal que los números pueden progresar alternativamente de manera positiva y negativa en la misma secuencia.

Preguntas frecuentes (FAQ) sobre la secuencia aritmética y geométrica

¿Por qué se llama secuencia geométrica?

Se llama secuencia geométrica porque los números van de un número a otro al bucear o multiplicar por un valor similar.

The number divided or multiplied at every stage of the series called the common ratio. A geometric series is a set of figures that follow a unique rule of a pattern.

¿Puede una secuencia aritmética ser también geométrica?

En matemáticas, una serie aritmética se define como la secuencia en la que la varianza entre números consecutivos llamada diferencia común es constante.

On the other hand, the geometric series is where the ratio between successive numbers, known as a common ratio, is constant. So, that means a sequence can’t be both geometric and arithmetic.

¿Cuál es la fórmula de la serie geométrica infinita?

The infinite geometric sequence is defined as a totality of an infinite geometric sequence. The sequence doesn’t have the last figure. This type of an infinite sequence include a1+a1r+a1r2 +a1r3+…. In this case, a1 refers to the first figure while r refers to the common ratio.

You will calculate the total sum of a finite geometric sequence. In the case of the infinite geometric sequence, once the common ratio is above one, the terms in the series will increase, and when you add larger numbers, getting a final answer will be impossible. The only answer would be infinity.

Let’s say the r (common ratio) lies between -1 and 1/. You can get the sum of an infinite geometric sequence. That’s, the sum exists for r <1.

La suma de series geométricas infinitas que tiene -1 S = a1 / 1-r

¿Qué es A en una secuencia aritmética?

Una secuencia aritmética se refiere a la serie de términos de tal manera que una diferencia entre dos participantes sucesivos de la serie es un término constante en el que a en la secuencia aritmética es el primer término.

¿Cómo hallas el enésimo término de una secuencia aritmética?

Se sabe que los términos de una serie aritmética aumentan en la diferencia común (d). Por ejemplo, 2, 4, 6, 8, 10 es una progresión aritmética yd = 2.

La fórmula para obtener el enésimo término de esta secuencia aritmética es 2n + 1. Normalmente, el enésimo término de una secuencia aritmética con un primer término y una diferencia común es a + (n-1) d.

Conclusión

With the help of this detailed discussion about the differences between an arithmetic sequence and a geometric sequence, you should be clear about it by now. If you think that these 2 sequences do not have any real-life uses, then you should think again. Both have their individual uses and importance in different day to day lives.

Arithmetic sequences are used in various financial sectors and can prove to be rather useful when it comes to calculating your savings and personal financial increments. However, a geometric sequence also has its fair share of uses. It is used to calculate interest rates provided by different financial institutions and also to calculate the population growth of a country.

It is often seen that students get confused when it comes to deciding whether a given sequence is an arithmetic sequence or a geometric sequence. Although calculating an arithmetic sequence is pretty simple, the main challenge lies in calculating a geometric sequence.

Referencias

  1. https://arxiv.org/pdf/1001.5055
  2. https://msp.org/pjm/1971/38-2/pjm-v38-n2-p05-s.pdf

1 comentario en “Difference Between Arithmetic and Geometric Sequence (With Table)”

  1. Soy profesor de matemáticas y me encanta resolver ecuaciones. Vine aquí buscando aritmética vs secuencia geométrica. La diferencia común y la razón común se han escrito para que cualquiera pueda entender. Buen trabajo.

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