Name: proving number patterns with induction
Uploaded: 2025-11-26T05:43:51.000Z
Duration: 1 min 29 s
Channel: JoVE

L’induction mathématique est une technique utilisée pour établir la vérité d’un énoncé pour tous les nombres naturels. Par exemple, dans un effet domino, si le premier tombe et que chacun renverse le suivant, toute la ligne tombera.Prenons un autre exemple d’épargne quotidienne : 1 trimestre le premier jour, 2 le deuxième, 3 le troisième, et ainsi de suite, jusqu’à n jours. Le nombre total de quarts forme une série qui comprend les n premiers nombres naturels. La somme de cette série suit un schéma simple qui, lorsqu’il est évalué, est égal à n fois n plus 1 sur 2.L’induction mathématique prouve ce modèle pour tous les nombres naturels en confirmant le cas de base et l’étape inductive.Pour le premier nombre, le cas de base, la somme réelle correspond au résultat de la règle, qui vérifie le cas de base.On suppose alors que la règle fonctionne pour n’importe quel nombre k. Cela forme l’étape inductive, où la somme jusqu’à ce nombre est censée suivre le même modèle.L’ajout du nombre suivant, k plus 1, est montré pour maintenir le modèle, en suivant la même logique.Avec le cas de base et l’étape inductive qui restent vrais, la règle est validée pour tous les nombres naturels par induction mathématique.

raisonnement par récurrence

Mathematical Induction

Le raisonnement par r&#233;currence est une m&#233;thode de preuve structur&#233;e employ&#233;e pour &#233;tablir la validit&#233; d&#8217;&#233;nonc&#233;s portant sur les nombres naturels. Consid&#233;rons la somme des n premiers nombres naturels&#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:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40096/40096_Equation_1_1__editor_40096.svg' alt='40096_Equation_1_1_.svg' /></figure>Cette formule d&#233;crit une propri&#233;t&#233; qui semble se maintenir lorsque l&#8217;on ajoute des termes. Pour v&#233;rifier qu&#8217;elle est valable pour tous les nombres naturels, le raisonnement par r&#233;currence se d&#233;roule en deux &#233;tapes essentielles. La premi&#232;re est l&#8217;&#233;tape d&#8217;initialisation, o&#249; la formule est test&#233;e pour la valeur initiale, g&#233;n&#233;ralement n = 1. En substituant dans les deux membres de l&#8217;&#233;galit&#233;, on confirme l&#8217;&#233;galit&#233; :<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40096/40096_Equation_2_1__editor_40096.svg' alt='40096_Equation_2_1_.svg' /></figure>Ainsi, l&#8217;&#233;tape d&#8217;initialisation est v&#233;rifi&#233;e. Vient ensuite l&#8217;&#233;tape d&#8217;h&#233;r&#233;dit&#233;, o&#249; l&#8217;on suppose la formule vraie pour un entier naturel arbitraire k. Cette supposition constitue l&#8217;hypoth&#232;se de r&#233;currence :<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40096/40096_Equation_3_1__editor_40096.svg' alt='40096_Equation_3_1_.svg' /></figure>On montre alors que la formule vaut aussi pour k + 1. En ajoutant k + 1 aux deux membres de l&#8217;hypoth&#232;se de r&#233;currence, on obtient&#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:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40096/40096_Equation_4_1__editor_40096.svg' alt='40096_Equation_4_1_.svg' /></figure>Cela confirme que la propri&#233;t&#233; est v&#233;rifi&#233;e pour k + 1. L&#8217;&#233;tape d&#8217;initialisation et l&#8217;&#233;tape d&#8217;h&#233;r&#233;dit&#233; &#233;tant &#233;tablies, la formule est d&#233;montr&#233;e pour tous les nombres naturels.

Raisonnement par récurrence

Le raisonnement par r&#233;currence est une m&#233;thode de preuve structur&#233;e employ&#233;e pour &#233;tablir la validit&#233; ...

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

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 inductive step. Here, the statement is assumed to be true for an arbitrary natural number k. This assumption is called the induction hypothesis. The goal is then to show that the same statement is true for k + 1.

To do that, k + 1 is added to both sides of the induction hypothesis. This shows that the pattern also works for the next number. With the base case and the inductive step both verified, the formula is proven for all natural numbers.

Proving Number Patterns with Induction

sequences

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...

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

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 mathematical induction proves number patterns using a base case and inductive step, with the sum of the first n natural numbers as an example.

Proving Number Patterns with Induction | 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

Proving Number Patterns with Induction

Description

Transcript

Tags