Name: infinite geometric series and decay
Uploaded: 2025-11-26T05:43:52.000Z
Duration: 1 min 30 s
Channel: JoVE

Considérez une balle tombée d’une certaine hauteur sur une surface dure. Après chaque rebond, il rebondit à une fraction fixe de la hauteur qu’il a atteinte lors du rebond précédent.Les hauteurs de rebond suivent un motif géométrique, chaque nouvelle hauteur étant une fraction cohérente de la précédente.La valeur fixe utilisée pour la multiplication est appelée rapport commun.Le premier terme représente la hauteur de rebond initiale, et chaque hauteur suivante est calculée en multipliant le terme précédent par le rapport commun.Ce processus modélise la décroissance exponentielle, car la balle perd de l’énergie à chaque rebond.Par la suite, la distance cumulée totale parcourue après plusieurs rebonds est calculée en doublant chaque hauteur de rebond pour tenir compte à la fois de la montée et de la descente, à l’exception de la chute initiale.Une fois que chaque terme de la séquence est doublé, des sommes partielles, définies comme la somme d’un nombre fixe de termes, peuvent être utilisées pour calculer la distance cumulative totale parcourue par la balle en tout point.Pour trouver la énième somme partielle, écrivez la somme et multipliez-la par le rapport, en décalant tous les termes vers l’avant. La soustraction annule les termes intermédiaires, ne laissant que le premier et le dernier.La factorisation et la division donnent la énième somme partielle, montrant comment les suites géométriques fonctionnent dans des situations réelles.

suites géométriques

Geometric Sequences

Dans les syst&#232;mes o&#249; les valeurs diminuent selon une proportion constante &#224; chaque &#233;tape, la suite r&#233;sultante pr&#233;sente une structure g&#233;om&#233;trique. Chaque nouvelle valeur de la suite s&#8217;obtient en appliquant un multiplicateur fixe au terme pr&#233;c&#233;dent. Ce type de d&#233;clin r&#233;gulier et proportionnel est souvent utilis&#233; pour repr&#233;senter des processus impliquant une perte progressive, comme la dissipation d&#8217;&#233;nergie ou la r&#233;duction de l&#8217;amplitude au fil du temps.Lorsqu'on analyse l'effet total d'un tel processus sur un nombre infini d'it&#233;rations, la suite de valeurs est appel&#233;e suite g&#233;om&#233;trique infinie. Supposons que la valeur du multiplicateur commun (ou raison) soit inf&#233;rieure &#224; un. Dans ce cas, les valeurs de la suite continuent de d&#233;cro&#238;tre et leur somme tend vers un nombre fini. Cette condition garantit la convergence plut&#244;t qu'une croissance ou une fluctuation illimit&#233;e. La formule permettant de calculer la somme totale d'une suite g&#233;om&#233;trique infinie est&#160;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:19.4%;'><img style='aspect-ratio:289/153;' src='https://cloudfront.jove.com/files/ftp_upload/40095/40095_Equation_1_editor_40095.svg' alt='40095_Equation_1.svg' /></figure>o&#249; S repr&#233;sente la somme de la s&#233;rie, a d&#233;signe le premier terme et r la raison. Cette expression math&#233;matique permet de calculer efficacement les r&#233;sultats &#224; long terme pour des processus caract&#233;ris&#233;s par une d&#233;croissance exponentielle. Ses applications couvrent divers domaines, notamment la physique, l&#8217;&#233;conomie et l&#8217;ing&#233;nierie, o&#249; de tels mod&#232;les de d&#233;croissance sont essentiels &#224; une pr&#233;diction et une analyse pr&#233;cises.

Suites géométriques

Dans les syst&#232;mes o&#249; les valeurs diminuent selon une proportion constante &#224; chaque &#233;tape, la suite r&#233;sultante pr&#233;sente une ...

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Math Fundamentals

Fondamentaux des mathématiques

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Introduction to Sequences and Series

Introduction aux suites et séries

KEY TERMS AND CONCEPTS

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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 called convergence. The sum of an infinite geometric series can be found with a formula that uses S for the sum, a for the first term, and r for the common ratio.

This method gives a fast way to calculate long-term results in processes with exponential decrease. It is used in physics, economics, and engineering, where decay models help with prediction and analysis.

Infinite Geometric Series and Decay

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 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 infinite geometric series model repeated decay, when they converge, and how the sum formula helps solve real-world problems in science and math.

Infinite Geometric Series and Decay | 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

Infinite Geometric Series and Decay

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