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

Ciąg arytmetyczny to lista liczb, w której każdy wyraz zwiększa się lub zmniejsza o tę samą stałą liczbę, znaną jako wspólna różnica. Weźmy pod uwagę stos słupów. Pierwsza warstwa zawiera 25 biegunów, a liczba biegunów zmniejsza się o 1 w każdej kolejnej warstwie.Biorąc pod uwagę, że stos ma 12 warstw, celem jest znalezienie całkowitej liczby słupów.Ten układ tworzy ciąg arytmetyczny, ponieważ liczba biegunów zmniejsza się o stałą wartość z jednej warstwy na drugą.W tym scenariuszu liczba biegunów w 12. warstwie jest obliczana przy użyciu wzoru na n-ty składnik ciągu arytmetycznego, w oparciu o pierwszy składnik, wspólną różnicę i liczbę warstw. Wartości tych terminów są następnie podstawiane do wzoru, który upraszcza się do 25 minus 11, co daje 14 biegunów w 12. warstwie.Całkowita liczba słupów w stosie, znana jako częściowa suma sekwencji, jest następnie obliczana, biorąc średnią liczby słupów w pierwszej i ostatniej warstwie i mnożąc ją przez całkowitą liczbę warstw. Nazywa się to sumą częściową, ponieważ dodawanych jest tylko pierwszych 12 wyrazów sekwencji. Daje to częściową sumę 12 pomnożoną przez średnią 25 i 14, co daje 234 bieguny.

ciągi arytmetyczne

Arithmetic Sequences

Ci&#261;g arytmetyczny to uporz&#261;dkowany ci&#261;g liczb, w kt&#243;rym ka&#380;dy wyraz wyznacza si&#281; przez dodanie sta&#322;ej warto&#347;ci, zwanej wsp&#243;ln&#261; r&#243;&#380;nic&#261;, do wyrazu poprzedniego. Ta sta&#322;a regu&#322;a umo&#380;liwia efektywne wyznaczanie dowolnego wyrazu ci&#261;gu oraz sumy cz&#281;&#347;ciowej wielu wyraz&#243;w. Wz&#243;r na znalezienie n-tego wyrazu ci&#261;gu arytmetycznego to:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:85.72%;'><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>W tym przypadku a_n oznacza n-ty wyraz ci&#261;gu, a jest pierwszym wyrazem, d jest wsp&#243;ln&#261; r&#243;&#380;nic&#261;, a n to numer (pozycja) wyrazu w ci&#261;gu. To r&#243;wnanie jest niezb&#281;dne do wyznaczenia warto&#347;ci dowolnego wyrazu bez konieczno&#347;ci wypisywania wszystkich poprzednich. Aby obliczy&#263; sum&#281; pierwszych n wyraz&#243;w, zwan&#261; sum&#261; cz&#281;&#347;ciow&#261;, stosuje si&#281; jeden z nast&#281;puj&#261;cych wzor&#243;w:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><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>W tych wyra&#380;eniach S_n oznacza sum&#281; pierwszych n wyraz&#243;w, a a_n ponownie odnosi si&#281; do n-tego wyrazu, wyznaczonego za pomoc&#261; wcze&#347;niejszego wzoru. Wzory te stanowi&#261; zwi&#281;z&#322;e i systematyczne narz&#281;dzie do analizy regularnych wzorc&#243;w liczbowych zar&#243;wno w zastosowaniach teoretycznych, jak i praktycznych.

Ciągi arytmetyczne

Ci&#261;g arytmetyczny to uporz&#261;dkowany ci&#261;g liczb, w kt&#243;rym ka&#380;dy wyraz wyznacza si&#281; przez dodanie sta&#322;ej warto&#347;ci, ...

Education

JoVE Core

Math Fundamentals

Podstawy matematyki

Unit

Jednostka

Introduction to Sequences and Series

Wprowadzenie do ciągów i szeregów

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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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