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

As sequências são listas ordenadas de números organizados de acordo com uma regra ou padrão específico. O enésimo termo é encontrado usando uma fórmula baseada em sua posição.Por exemplo, a altura de uma bola quicando diminui a cada salto, formando uma sequência decrescente em que cada altura é uma fração fixa da anterior.Cada número em uma sequência é chamado de termo, e a posição dos termos ordenados determina seu valor.Quando o padrão é claro, os pontos indicam que a sequência continua.Algumas sequências definem termos usando os anteriores, chamados de sequências recursivas. Por exemplo, o enésimo termo é definido usando o (n-1)ésimo termo.Por exemplo, na sequência de Fibonacci, cada termo é igual à soma dos dois termos anteriores.Somas parciais são as somas dos primeiros termos de uma sequência. Freqüentemente mostradas usando a notação sigma, as somas parciais ajudam a analisar como a soma cresce à medida que mais termos são adicionados.Cada um deles é chamado de soma parcial: S1 é o primeiro, S2 é o segundo e Sn é a soma do termo n-ésimo.A sequência formada por eles é chamada de sequência de somas parciais.Por exemplo, o depósito de cada semana é um termo ao rastrear economias semanais. As somas parciais mostram como a poupança total cresce ao longo do tempo.

sequências

Sequences

Sequ&#234;ncias s&#227;o objetos matem&#225;ticos fundamentais que consistem em listas ordenadas de n&#250;meros que obedecem a uma regra ou padr&#227;o espec&#237;fico. Sequ&#234;ncias s&#227;o essenciais em diversos conceitos matem&#225;ticos, incluindo c&#225;lculo, s&#233;ries e teoria dos n&#250;meros. Elas podem modelar fen&#244;menos do mundo real, como crescimento populacional, investimentos financeiros e processos f&#237;sicos, como a diminui&#231;&#227;o da altura de uma bola quicando.Cada n&#250;mero em uma sequ&#234;ncia &#233; chamado de termo. Normalmente, os termos s&#227;o denotados como a_1, a_2, a_3,&#8230;, onde o subscrito indica a posi&#231;&#227;o dentro da sequ&#234;ncia. Quando o padr&#227;o &#233; evidente, as sequ&#234;ncias geralmente incluem retic&#234;ncias para indicar continua&#231;&#227;o.Matematicamente, uma sequ&#234;ncia tamb&#233;m pode ser vista como uma fun&#231;&#227;o cujo dom&#237;nio &#233; o conjunto dos n&#250;meros naturais, com cada n&#250;mero natural mapeado para um termo espec&#237;fico. Essa representa&#231;&#227;o funcional &#233; &#250;til ao definir sequ&#234;ncias de forma expl&#237;cita ou recursiva.Sequ&#234;ncias recursivas definem cada termo com base em termos precedentes. Um dos exemplos mais conhecidos &#233; a sequ&#234;ncia de Fibonacci, na qual cada termo &#233; igual &#224; soma dos dois termos precedentes:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:35.58%;'><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>Essa defini&#231;&#227;o enfatiza a depend&#234;ncia de cada termo em rela&#231;&#227;o aos seus predecessores, caracter&#237;stica das sequ&#234;ncias recursivas.Somas parciais s&#227;o as somas dos primeiros termos de uma sequ&#234;ncia e s&#227;o &#250;teis para analisar como a soma cumulativa evolui &#224; medida que mais termos s&#227;o adicionados. Para uma sequ&#234;ncia {a_n}, a n^a soma parcial &#233; dada por:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:35.58%;'><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>Compreender sequ&#234;ncias, defini&#231;&#245;es recursivas e somas parciais constitui a base para explorar t&#243;picos mais avan&#231;ados, como s&#233;ries infinitas, converg&#234;ncia e indu&#231;&#227;o matem&#225;tica.Al&#233;m disso, uma sequ&#234;ncia telesc&#243;pica &#233; um tipo especial de sequ&#234;ncia em que a maioria dos termos se cancela quando a soma parcial &#233; expandida, facilitando a avalia&#231;&#227;o da soma. Em uma s&#233;rie telesc&#243;pica, a n-&#233;sima soma parcial frequentemente se reduz &#224; diferen&#231;a entre alguns termos, normalmente o primeiro e o &#250;ltimo, permitindo uma express&#227;o em forma fechada simplificada. Essa propriedade &#233; especialmente &#250;til na avalia&#231;&#227;o de s&#233;ries infinitas e na demonstra&#231;&#227;o de converg&#234;ncia.

Sequências

Sequ&#234;ncias s&#227;o objetos matem&#225;ticos fundamentais que consistem em listas ordenadas de n&#250;meros que obedecem a uma regra ou padr&#227;o ...

Education

JoVE Core

Math Fundamentals

Fundamentos da Matemática

Unit

Unidade

Introduction to Sequences and Series

Introdução a Sequências e Séries

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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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