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

Les séquences sont des listes ordonnées de nombres disposées selon une règle ou un modèle spécifique. Le nième terme est trouvé à l’aide d’une formule basée sur sa position.Par exemple, la hauteur d’une balle rebondissante diminue à chaque rebond, formant une séquence décroissante dans laquelle chaque hauteur est une fraction fixe de la précédente.Chaque nombre d’une suite est appelé un terme, et la position des termes ordonnés détermine sa valeur.Lorsque le motif est clair, des points indiquent que la séquence se poursuit.Certaines séquences définissent des termes à l’aide des précédents, appelées suites récursives. Par exemple, le nième terme est défini à l’aide du (n-1)ème terme.Par exemple, dans la suite de Fibonacci, chaque terme est égal à la somme des deux termes précédents.Les sommes partielles sont les sommes des premiers termes d’une suite. Souvent illustrées à l’aide de la notation sigma, les sommes partielles permettent d’analyser comment la somme augmente à mesure que d’autres termes sont ajoutés.Chacun d’entre eux est appelé une somme partielle : S1 est le premier, S2 est le second et Sn est la sommation des termes du nième.La suite formée par ceux-ci est appelée la suite des sommes partielles.Par exemple, le dépôt de chaque semaine est un terme pour le suivi de l’épargne hebdomadaire. Des sommes partielles montrent comment l’épargne totale augmente au fil du temps.

suites

Sequences

Les suites sont des objets math&#233;matiques fondamentaux constitu&#233;s de listes ordonn&#233;es de nombres suivant une r&#232;gle ou loi de formation sp&#233;cifique. Elles sont essentielles pour divers concepts math&#233;matiques, notamment l&#8217;analyse, les s&#233;ries et la th&#233;orie des nombres. Elles permettent de mod&#233;liser des ph&#233;nom&#232;nes r&#233;els tels que la croissance d&#233;mographique, les investissements financiers et des processus physiques comme la d&#233;croissance de la hauteur de rebond d&#8217;une balle.Chaque nombre d&#8217;une suite est appel&#233; un terme. G&#233;n&#233;ralement, les termes sont not&#233;s a_1, a_2, a_3, &#8230;, o&#249; l&#8217;indice d&#233;signe la position dans la suite. Lorsque la r&#233;gularit&#233; est manifeste, on recourt souvent aux points de suspension pour indiquer la suite.Math&#233;matiquement, une suite peut &#233;galement &#234;tre consid&#233;r&#233;e comme une fonction dont le domaine est l&#8217;ensemble des entiers naturels, chaque entier naturel &#233;tant associ&#233; &#224; un terme sp&#233;cifique. Cette repr&#233;sentation fonctionnelle est utile pour d&#233;finir des suites de mani&#232;re explicite ou r&#233;cursive.Les suites r&#233;cursives d&#233;finissent chaque terme en fonction des termes pr&#233;c&#233;dents. L&#8217;un des exemples les plus c&#233;l&#232;bres est la suite de Fibonacci, o&#249; chaque terme est &#233;gal &#224; la somme des deux termes pr&#233;c&#233;dents&#160;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><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>Cette d&#233;finition souligne la d&#233;pendance de chaque terme &#224; l&#8217;&#233;gard de ses pr&#233;d&#233;cesseurs, caract&#233;ristique des suites r&#233;cursives.Les sommes partielles sont les sommes des premiers termes d&#8217;une suite et sont utiles pour analyser l&#8217;&#233;volution de la somme cumul&#233;e &#224; mesure que des termes sont ajout&#233;s. Pour une suite {a_n}, la n-i&#232;me somme partielle est donn&#233;e par&#160;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><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>La compr&#233;hension des suites, des d&#233;finitions r&#233;cursives et des sommes partielles constitue la base de l&#8217;exploration de sujets plus avanc&#233;s tels que les s&#233;ries infinies, la convergence et l&#8217;induction math&#233;matique.De plus, une suite t&#233;lescopique est un type particulier de suite dans laquelle la plupart des termes s&#8217;annulent lorsque la somme partielle est d&#233;velopp&#233;e, ce qui en facilite l&#8217;&#233;valuation. Dans une s&#233;rie t&#233;lescopique, la n-i&#232;me somme partielle se r&#233;duit souvent &#224; la diff&#233;rence entre quelques termes, g&#233;n&#233;ralement le premier et le dernier, ce qui permet d&#8217;obtenir une expression ferm&#233;e simplifi&#233;e. Cette propri&#233;t&#233; est particuli&#232;rement utile pour &#233;valuer les s&#233;ries infinies et d&#233;montrer la convergence.

Suites

Les suites sont des objets math&#233;matiques fondamentaux constitu&#233;s de listes ordonn&#233;es de nombres suivant une r&#232;gle ou loi de formation ...

Education

Enseignement

JoVE Core

Math Fundamentals

Fondamentaux des mathématiques

Unit

Unité

Introduction to Sequences and Series

Introduction aux suites et séries

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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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