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

数学的帰納法は、すべての自然数のステートメントの真実性を確立するために使用される手法です。たとえば、ドミノ効果では、最初のものが倒れ、それぞれが次のものが倒れると、線全体が倒れます。毎日四半期を貯蓄する別の例を考えてみましょう (初日は 1 四半期、2 日目は 2 四半期、3 日目は 3 日、というように、 最大 n 日まで)。四半期の総数は、最初の n 個の自然数を含む系列を形成します。この系列の合計は、評価すると、n x に n を加えた 1 を 2 に等しい単純なパターンに従います。数学的帰納法は、基本ケースと帰納ステップを確認することにより、すべての自然数についてこのパターンを証明します。最初の数値である基本ケースの場合、実際の合計は、基本ケースを検証するルールの結果と一致します。次に、ルールが任意の数値 kに対して機能すると仮定します。これにより帰納ステップが形成され、その数までの合計が同じパターンに従うことが期待されます。次の数値 k プラス 1 を加算すると、同じロジックに従ってパターンが維持されます。基本ケースと帰納的ステップの両方が当てはまると、法則は数学的帰納法によってすべての自然数に対して検証されます。

数学的帰納法

Mathematical Induction

&#25968;&#23398;&#30340;&#24112;&#32013;&#27861;&#12399;&#12289;&#33258;&#28982;&#25968;&#12395;&#38306;&#12377;&#12427;&#21629;&#38988;&#12398;&#27491;&#12375;&#12373;&#12434;&#20307;&#31995;&#30340;&#12395;&#35388;&#26126;&#12377;&#12427;&#12383;&#12417;&#12398;&#26041;&#27861;&#12391;&#12377;&#12290;&#12383;&#12392;&#12360;&#12400;&#12289;&#26368;&#21021;&#12398;n&#20491;&#12398;&#33258;&#28982;&#25968;&#12398;&#21644;&#12434;&#32771;&#12360;&#12414;&#12377;&#12290;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:45.39%;'><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>&#12371;&#12398;&#24335;&#12399;&#12289;&#38917;&#12434;&#21152;&#12360;&#12390;&#12356;&#12367;&#12392;&#24120;&#12395;&#25104;&#12426;&#31435;&#12388;&#12424;&#12358;&#12395;&#35211;&#12360;&#12427;&#31561;&#24335;&#12398;&#38306;&#20418;&#12434;&#34920;&#12375;&#12390;&#12356;&#12414;&#12377;&#12290;&#12371;&#12398;&#24335;&#12364;&#12377;&#12409;&#12390;&#12398;&#33258;&#28982;&#25968;n&#12395;&#23550;&#12375;&#12390;&#25104;&#12426;&#31435;&#12388;&#12371;&#12392;&#12434;&#31034;&#12377;&#12383;&#12417;&#12395;&#12289;&#25968;&#23398;&#30340;&#24112;&#32013;&#27861;&#12399;&#27425;&#12398;&#20108;&#12388;&#12398;&#20027;&#35201;&#12394;&#27573;&#38542;&#12391;&#36914;&#12417;&#12425;&#12428;&#12414;&#12377;&#12290; &#26368;&#21021;&#12398;&#27573;&#38542;&#12399;&#22522;&#24213;&#12398;&#22580;&#21512;&#12391;&#12377;&#12290;&#12371;&#12371;&#12391;&#12399;&#12289;&#36890;&#24120;n = 1&#12398;&#12392;&#12365;&#12395;&#24335;&#12364;&#25104;&#12426;&#31435;&#12388;&#12363;&#12393;&#12358;&#12363;&#12434;&#30906;&#35469;&#12375;&#12414;&#12377;&#12290;&#20001;&#36794;&#12395;n = 1&#12434;&#20195;&#20837;&#12377;&#12427;&#12392;&#12289;&#31561;&#24335;&#12364;&#25104;&#31435;&#12377;&#12427;&#12371;&#12392;&#12364;&#30906;&#12363;&#12417;&#12425;&#12428;&#12414;&#12377;&#12290;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:43.05%;'><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>&#12375;&#12383;&#12364;&#12387;&#12390;&#12289;&#22522;&#24213;&#12398;&#22580;&#21512;&#12364;&#25104;&#31435;&#12375;&#12414;&#12377;&#12290;&#27425;&#12398;&#27573;&#38542;&#12399;&#24112;&#32013;&#27573;&#38542;&#12391;&#12377;&#12290;&#12371;&#12371;&#12391;&#12399;&#12289;&#20219;&#24847;&#12398;&#33258;&#28982;&#25968;k&#12395;&#23550;&#12375;&#12390;&#24335;&#12364;&#25104;&#12426;&#31435;&#12388;&#12392;&#20206;&#23450;&#12375;&#12414;&#12377;&#12290;&#12371;&#12398;&#20206;&#23450;&#12434;&#12300;&#24112;&#32013;&#27861;&#12398;&#20206;&#23450;&#12301;&#12392;&#21628;&#12403;&#12414;&#12377;&#12290;<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>&#27425;&#12395;&#12289;&#12371;&#12398;&#24335;&#12364;k+1&#12398;&#22580;&#21512;&#12395;&#12418;&#25104;&#12426;&#31435;&#12388;&#12371;&#12392;&#12434;&#31034;&#12375;&#12414;&#12377;&#12290;&#24112;&#32013;&#27861;&#12398;&#20206;&#23450;&#12398;&#20001;&#36794;&#12395;k+1&#12434;&#21152;&#12360;&#12427;&#12392;&#12289;&#27425;&#12398;&#12424;&#12358;&#12395;&#12394;&#12426;&#12414;&#12377;&#12290;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:48.93%;'><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>&#12371;&#12398;&#32080;&#26524;&#12289;&#24335;&#12364;k+1&#12395;&#23550;&#12375;&#12390;&#12418;&#25104;&#31435;&#12377;&#12427;&#12371;&#12392;&#12364;&#30906;&#35469;&#12373;&#12428;&#12414;&#12377;&#12290;&#22522;&#24213;&#12398;&#22580;&#21512;&#12392;&#24112;&#32013;&#27573;&#38542;&#12398;&#20001;&#26041;&#12364;&#25104;&#12426;&#31435;&#12388;&#12371;&#12392;&#12391;&#12289;&#12371;&#12398;&#24335;&#12399;&#12377;&#12409;&#12390;&#12398;&#33258;&#28982;&#25968;&#12395;&#12388;&#12356;&#12390;&#30495;&#12391;&#12354;&#12427;&#12371;&#12392;&#12364;&#35388;&#26126;&#12373;&#12428;&#12414;&#12377;&#12290;

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

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