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

假设将一个球从某一高度 dropped 到坚硬的表面上。每次弹起后，它都会反弹到前一次弹起高度的固定比例。回弹高度遵循几何模式，每一次新的高度均为前一次高度的一个固定比例。用于乘法的固定值称为公比。第一项表示初始反弹高度，每一后续高度通过将前一项乘以公比来计算。此过程模拟了指数衰减，因为球在每次弹跳时都会损失能量。随后，通过将每次反弹高度乘以 2 来计算多次反弹后的总累计距离，以同时考虑上升和下降过程，但初始下落除外。当序列中的每一项都加倍后，可通过部分和——即固定数量项的和——来计算小球在任意时刻所经过的总累计距离。要求得第 n 个部分和，可先写出该和式，再将其乘以公比，使所有项向前移动一位。相减后中间项相互抵消，仅剩下首项和末项。因式分解与除法运算得到第 n 项部分和，展示了几何数列在实际情境中的工作原理。

几何序列

Geometric Sequences

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&#24403;&#20998;&#26512;&#27492;&#31867;&#36807;&#31243;&#32463;&#36807;&#26080;&#38480;&#27425;&#36845;&#20195;&#21518;&#30340;&#24635;&#20307;&#25928;&#24212;&#26102;&#65292;&#35813;&#24207;&#21015;&#34987;&#31216;&#20026;&#26080;&#38480;&#20960;&#20309;&#32423;&#25968;&#12290;&#33509;&#20844;&#27604;&#30340;&#32477;&#23545;&#20540;&#23567;&#20110; 1&#65292;&#21017;&#24207;&#21015;&#20013;&#30340;&#21508;&#39033;&#23558;&#19981;&#26029;&#20943;&#23567;&#65292;&#20854;&#32047;&#35745;&#21644;&#36235;&#36817;&#20110;&#19968;&#20010;&#30830;&#23450;&#30340;&#26377;&#38480;&#20540;&#12290;&#36825;&#31181;&#26465;&#20214;&#20445;&#35777;&#32423;&#25968;&#30340;&#25910;&#25947;&#65292;&#32780;&#38750;&#26080;&#30028;&#22686;&#38271;&#25110;&#25391;&#33633;&#12290;
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40095_Equation_1.svg
&#20854;&#20013;&#65292;S &#34920;&#31034;&#32423;&#25968;&#30340;&#24635;&#21644;&#65292;a &#34920;&#31034;&#21021;&#22987;&#20540;&#65292;r &#34920;&#31034;&#20844;&#27604;&#12290;&#35813;&#25968;&#23398;&#34920;&#36798;&#24335;&#33021;&#22815;&#39640;&#25928;&#22320;&#35745;&#31639;&#21576;&#25351;&#25968;&#36882;&#20943;&#29305;&#24449;&#30340;&#36807;&#31243;&#30340;&#38271;&#26399;&#32467;&#26524;&#12290;&#27492;&#31867;&#27169;&#22411;&#24191;&#27867;&#24212;&#29992;&#20110;&#29289;&#29702;&#23398;&#12289;&#32463;&#27982;&#23398;&#19982;&#24037;&#31243;&#23398;&#31561;&#39046;&#22495;&#65292;&#23545;&#31934;&#30830;&#39044;&#27979;&#19982;&#20998;&#26512;&#20855;&#26377;&#37325;&#35201;&#24847;&#20041;&#12290;

Education

教育

JoVE Core

Math Fundamentals

数学基础

Unit

单位

Introduction to Sequences and Series

数列与级数导论

KEY TERMS AND CONCEPTS

关键术语与概念

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

Description

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