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

シーケンスは、特定のルールまたはパターンに従って配置された数値の順序付きリストです。n番目の項は、その位置に基づく式を使用して検索されます。たとえば、跳ね返るボールの高さは跳ね返るたびに減少し、各高さが前の高さの固定部分である減少シーケンスを形成します。シーケンス内の各数値は項と呼ばれ、順序付けられた項の位置によってその値が決まります。パターンが明確な場合、ドットはシーケンスが続行されることを示します。一部のシーケンスは、再帰シーケンスと呼ばれる以前の用語を使用して用語を定義します。たとえば、n 番目の項は (n-1) 番目の項を使用して定義されます。たとえば、フィボナッチ数列では、各項は前の 2 つの項の合計に等しくなります。部分合計は、シーケンスの最初の数項の合計です。多くの場合、シグマ表記を使用して表示される部分合計は、より多くの項が追加されるにつれて合計がどのように増加するかを分析するのに役立ちます。これらはそれぞれ部分和と呼ばれ、 S1 は最初の合計、 S2 は2番目の合計、 Sn は n番目の項の合計です。これらによって形成されるシーケンスは、部分和のシーケンスと呼ばれます。たとえば、毎週の預金は、毎週の貯蓄を追跡する際の用語です。部分的な合計は、総節約額が時間の経過とともにどのように増加するかを示します。

数列

Sequences

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Education

JoVE Core

Math Fundamentals

数学の基礎

Unit

ユニット

Introduction to Sequences and Series

数列と級数への導入

KEY TERMS AND CONCEPTS

主要用語と概念

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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