Name: arithmetic sequence formulas and partial sums
Uploaded: 2025-11-26T05:43:53.000Z
Duration: 1 min 30 s
Channel: JoVE

Eine arithmetische Folge ist eine Liste von Zahlen, bei der jeder Term um dieselbe feste Zahl zu- oder abnimmt, die als gemeinsame Differenz bezeichnet wird. Stellen Sie sich einen Haufen Stangen vor. Die erste Schicht enthält 25 Pole, und die Anzahl der Pole nimmt in jeder nachfolgenden Schicht um 1 ab.Da der Stapel aus 12 Schichten besteht, besteht das Ziel darin, die Gesamtzahl der Pole zu ermitteln.Diese Anordnung bildet eine arithmetische Sequenz, da die Anzahl der Pole von einer Schicht zur nächsten um einen konstanten Betrag abnimmt.In diesem Szenario wird die Anzahl der Pole in der 12. Schicht anhand der Formel für den n-ten Term einer arithmetischen Folge berechnet, basierend auf dem ersten Term, der gemeinsamen Differenz und der Anzahl der Layer. Die Werte dieser Terme werden dann in die Formel eingesetzt, die sich auf 25 minus 11 vereinfacht, was 14 Pole in der 12. Schicht ergibt.Die Gesamtzahl der Pole im Stapel, die als Teilsumme der Sequenz bezeichnet wird, wird dann berechnet, indem der Durchschnitt der Anzahl der Pole im ersten und letzten Layer mit der Gesamtzahl der Layer multipliziert wird. Sie wird als Teilsumme bezeichnet, da nur die ersten 12 Terme der Sequenz addiert werden. Daraus ergibt sich eine Teilsumme von 12 multipliziert mit dem Durchschnitt von 25 und 14, was 234 Pole ergibt.

arithmetische folgen

Arithmetic Sequences

Eine arithmetische Folge ist eine geordnete Zahlenfolge, bei der jedes aufeinanderfolgende Glied durch Addition einer Konstanten d (gemeinsame Differenz) zum vorhergehenden Glied entsteht. Dieses regelm&#228;&#223;ige Muster erm&#246;glicht sowohl die effiziente Berechnung beliebiger Folgeglieder als auch die Ermittlung von Summen mehrerer Glieder. Die Formel f&#252;r das n-te Glied einer arithmetischen Folge lautet:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:77.68%;'><img style='aspect-ratio:1920/74;' src='https://cloudfront.jove.com/files/ftp_upload/40094/40094_Equation_1_editor_40094.svg' alt='40094_Equation_1.svg' /></figure>Hier bezeichnet a_n das n-te Glied der Folge, a das erste Glied, d die gemeinsame Differenz und n den Positionsindex (Index) in der Folge. Diese Formel ist wesentlich, um den Wert eines beliebigen Glieds zu bestimmen, ohne alle vorangehenden Glieder aufzulisten. Zur Berechnung der Summe der ersten n Glieder (Teilsumme) verwendet man eine der folgenden Formeln:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:77.93%;'><img style='aspect-ratio:1920/163;' src='https://cloudfront.jove.com/files/ftp_upload/40094/40094_Equation_2_editor_40094.svg' alt='40094_Equation_2.svg' /></figure>In diesen Ausdr&#252;cken steht S_n f&#252;r die Summe der ersten n Glieder, und a_n bezeichnet erneut das n-te Glied, das mit der zuvor angegebenen Formel ermittelt wird. Diese Formeln bieten einen pr&#228;zisen und systematischen Ansatz zur Analyse regelm&#228;&#223;iger Zahlenmuster (mit konstanter Differenz) in theoretischen wie praktischen Anwendungen.

Arithmetische Folgen

Eine arithmetische Folge ist eine geordnete Zahlenfolge, bei der jedes aufeinanderfolgende Glied durch Addition einer Konstanten d (gemeinsame Differenz) ...

Education

Lehre

JoVE Core

Math Fundamentals

Mathematische Grundlagen

Unit

Einheit

Introduction to Sequences and Series

Einführung in Folgen und Reihen

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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.

The total of the first n terms is called the partial sum. To find it, you can use either of two formulas. In these expressions, S n means the sum of the first n terms, and a n is the nth term found from the earlier formula. These tools give a clear way to analyze evenly spaced numerical patterns in math and in practical situations.

Arithmetic Sequence Formulas and Partial Sums

arithmetic-sequence-formulas-and-partial-sums

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 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 arithmetic sequences use a common difference, the nth-term formula, and partial sum formulas to find terms and totals in a number pattern.

Arithmetic Sequence Formulas 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

Arithmetic Sequence Formulas and Partial Sums

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