Name: recursive sequences and partial sums
Uploaded: 2025-11-26T05:43:56.000Z
Duration: 1 min 29 s
Channel: JoVE

Las secuencias son listas ordenadas de números organizadas de acuerdo con una regla o patrón específico. El enésimo término se encuentra usando una fórmula basada en su posición.Por ejemplo, la altura de una pelota que rebota disminuye con cada rebote, formando una secuencia decreciente en la que cada altura es una fracción fija de la anterior.Cada número de una secuencia se denomina término y la posición de los términos ordenados determina su valor.Cuando el patrón es claro, los puntos indican que la secuencia continúa.Algunas secuencias definen términos usando los anteriores, llamados secuencias recursivas. Por ejemplo, el término n-ésimo se define utilizando el término (n-1).Por ejemplo, en la secuencia de Fibonacci, cada término es igual a la suma de los dos términos anteriores.Las sumas parciales son las sumas de los primeros términos de una secuencia. A menudo se muestran usando notación sigma, las sumas parciales ayudan a analizar cómo crece la suma a medida que se agregan más términos.Cada uno de estos se llama suma parcial: S1 es el primero, S2 es el segundo y Sn es la suma del término n-ésimo.La secuencia formada por estos se llama secuencia de sumas parciales.Por ejemplo, el depósito de cada semana es un plazo cuando se realiza un seguimiento de los ahorros semanales. Las sumas parciales muestran cómo crecen los ahorros totales con el tiempo.

sucesiones

Sequences

Las sucesiones son objetos matem&#225;ticos fundamentales que consisten en listas ordenadas de n&#250;meros que siguen una regla o un patr&#243;n espec&#237;fico. Son cruciales en diversos conceptos matem&#225;ticos, como el c&#225;lculo, las series y la teor&#237;a de n&#250;meros. Permiten modelar fen&#243;menos del mundo real como el crecimiento poblacional, las inversiones financieras y procesos f&#237;sicos como la disminuci&#243;n de la altura de una pelota que rebota.Cada n&#250;mero en una sucesi&#243;n se denomina t&#233;rmino. Habitualmente, los t&#233;rminos se denotan como a_1, a_2, a_3, &#8230;, donde el sub&#237;ndice indica la posici&#243;n dentro de la sucesi&#243;n. Cuando el patr&#243;n es evidente, se emplean puntos suspensivos para indicar su continuaci&#243;n.Desde un punto de vista matem&#225;tico, una sucesi&#243;n puede considerarse una funci&#243;n cuyo dominio es el conjunto de los n&#250;meros naturales, asignando a cada n&#250;mero natural un t&#233;rmino espec&#237;fico. Esta representaci&#243;n funcional es &#250;til al definir sucesiones de forma expl&#237;cita o recursiva.Las sucesiones recursivas definen cada t&#233;rmino en funci&#243;n de t&#233;rminos anteriores. Uno de los ejemplos m&#225;s famosos es la sucesi&#243;n de Fibonacci, donde cada t&#233;rmino es igual a la suma de los dos anteriores:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:35.18%;'><img style='aspect-ratio:1244/72;' src='https://cloudfront.jove.com/files/ftp_upload/40093/40093_Equation_1_editor_40093.svg' alt='40093_Equation_1.svg' /></figure>Esta definici&#243;n enfatiza la dependencia de cada t&#233;rmino con respecto a sus predecesores, lo cual es caracter&#237;stico de las sucesiones recursivas.Las sumas parciales son las sumas de los primeros pocos t&#233;rminos de una sucesi&#243;n y resultan valiosas para analizar c&#243;mo evoluciona la suma acumulada a medida que se a&#241;aden m&#225;s t&#233;rminos. Para una sucesi&#243;n {a_n}, la n-&#233;sima suma parcial se expresa como:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:34.39%;'><img style='aspect-ratio:1109/132;' src='https://cloudfront.jove.com/files/ftp_upload/40093/40093_Equation_2_editor_40093.svg' alt='40093_Equation_2.svg' /></figure>Comprender las sucesiones, las definiciones recursivas y las sumas parciales constituye la base fundamental para explorar temas m&#225;s avanzados, como las series infinitas, la convergencia y la inducci&#243;n matem&#225;tica.Adem&#225;s, una sucesi&#243;n telesc&#243;pica es un tipo especial de sucesi&#243;n en la que la mayor&#237;a de los t&#233;rminos se anulan al expandir la suma parcial, lo que facilita su c&#225;lculo. En una serie telesc&#243;pica, la n-&#233;sima suma parcial suele reducirse a la diferencia entre unos pocos t&#233;rminos, t&#237;picamente el primero y el &#250;ltimo, lo que permite una expresi&#243;n simplificada en forma cerrada.

Sucesiones

Las sucesiones son objetos matem&#225;ticos fundamentales que consisten en listas ordenadas de n&#250;meros que siguen una regla o un patr&#243;n ...

Education

Educación

JoVE Core

Math Fundamentals

Fundamentos de matemáticas

Unit

Unidad

Introduction to Sequences and Series

Introducción a las sucesiones y series

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

recursive sequences and partial sums

Sequences are ordered lists of numbers that follow a rule or pattern. Each number in the list is called a term. Terms are often written as a1, a2, a3, and so on, where the subscript shows the position in the sequence. When the pattern is clear, dots may be used to show that the list continues.

A sequence can also be viewed as a function. Its domain is the set of natural numbers, and each natural number maps to one term in the list. This view is useful when a sequence is defined explicitly or recursively. In a recursive sequence, each term depends on earlier terms. The Fibonacci sequence is a well-known example, because each term is the sum of the two terms before it.

Sequences appear in many areas of math, including calculus, series, and number theory. They can also model real-world changes such as population growth, financial investments, and the decreasing height of a bouncing ball. These examples show how a pattern in numbers can describe a changing process over time.

Partial sums help track the total of the first few terms of a sequence. For a sequence {an}, the nth partial sum is the sum of the first n terms. This idea is important for studying how a total changes as more terms are added. It also supports later topics such as infinite series, convergence, and mathematical induction.

A telescoping sequence is a special case where many terms cancel when the partial sum is expanded. This makes the sum easier to evaluate. In a telescoping series, the nth partial sum often becomes the difference between just a few terms, usually the first and the last. That simplified form is useful for finding closed-form expressions and for studying infinite series.

Recursive Sequences and Partial Sums

sequences

Sequences are ordered lists of numbers that follow a rule or pattern. Each number in the list is called a term. Terms are often written as a1, a2, a3, and so on, where the subscript shows the position in the sequence. When the pattern is clear, dots may be used to show that the list continues. A sequence can also be viewed as a function. Its domain is the set of natural numbers, and each natural number maps to one term in the list. This view is useful when a sequence is defined explicitly or...

summation notation

Summation Notation for Sequences

Sigma notation, also called summation notation, gives a short way to write the sum of a sequence of terms that follow a regular pattern. It uses the uppercase Greek letter sigma, ∑, to show that terms are being added together. A typical sigma expression includes an index of summation. The index starts at a lower value, such as 1, and ends at an upper value, such as n. The term a_k represents the general term in the sequence, where k changes across the stated range. For example, the increasing...

arithmetic sequences

Arithmetic Sequence Formulas and Partial Sums

Arithmetic sequences follow a regular number pattern. Each term is found by adding the same value, called the common difference, to the term before it. This steady change makes the sequence easy to study and predict. The nth-term formula helps find any term in the sequence without listing every earlier number. In this formula, a n is the nth term, a is the first term, d is the common difference, and n shows the term position. It gives a direct way to calculate a specific value from the pattern.

geometric sequences

Infinite Geometric Series and Decay

Infinite geometric series describe values that shrink by the same factor each step. Each new term is found by multiplying the previous term by a fixed common ratio. This pattern is useful for modeling gradual loss, such as energy dissipation or a drop in amplitude over time. When the common ratio has a magnitude less than one, the terms keep getting smaller. In that case, the total of the series approaches a definite number instead of growing without bound or changing back and forth. This is...

mathematical induction

Proving Number Patterns with Induction

Mathematical induction is a proof method for statements about natural numbers. It is often used to test whether a pattern really works for every number in a sequence. A common example is the sum of the first n natural numbers, where the formula seems to hold as more terms are added. The proof begins with a base case. In this step, the formula is checked at the first value, usually n = 1. Substituting 1 into both sides shows that the equality is true, so the base case holds. Next comes the...

the binomial theorem

Expanding Binomials with the Binomial Theorem

The Binomial Theorem gives a clear method for expanding binomial expressions raised to a power. It applies to binomials of the form (a + b)^n, where a and b are variables or constants that represent algebraic expressions, and n is a non-negative integer. Each term in the expansion follows a fixed pattern. The binomial coefficient in each term is found using factorials. The exponent of a decreases from n to 0, while the exponent of b increases from 0 to n. The theorem also helps students...

binomial expansion using pascal's triangle

binomial-expansion-using-pascal-s-triangle

Pascal’s Triangle for Binomial Coefficients

Pascal’s Triangle helps organize the coefficients in a binomial expansion. A binomial is an expression with two terms, such as (a + b), and raising it to a power creates a predictable pattern of terms. The triangle is built row by row. The top row, called the zeroth row, matches (a + b)0. Each new row gives the coefficients for the next power of the binomial, so the sixth row, 1, 5, 10, 10, 5, 1, gives the coefficients for (a + b)5. Each row begins and ends with 1. Every number inside the...

key terms and concepts

Learn how recursive sequences, partial sums, and telescoping series work, with examples like the Fibonacci sequence and real-world patterns.

Recursive Sequences and Partial Sums | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Recursive Sequences and Partial Sums

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