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

Consider a ball being dropped from a certain height onto a hard surface. After each bounce, it rebounds to a fixed fraction of the height it reached in the previous bounce.The rebound heights follow a geometric pattern, with each new height being a consistent fraction of the one before.The fixed value used for multiplication is called the common ratio.The first term represents the initial rebound height, and each subsequent height is calculated by multiplying the previous term by the common ratio.This process models exponential decay, as the ball loses energy with each bounce.Subsequently, the total cumulative distance covered after several bounces is calculated by doubling each rebound height to account for both the ascent and descent, except for the initial drop.Once each term of the sequence is doubled, partial sums&#8212;defined as the sum of a fixed number of terms&#8212;can be used to calculate the ball&#8217;s total cumulative distance covered at any point.To find the nth partial sum, write the sum and multiply it by the ratio, shifting all terms forward. Subtracting cancels the middle terms, leaving only the first and last.Factoring and dividing yield the nth partial sum, showing how geometric sequences work in real situations.

geometric sequences

Geometric Sequences

In systems where values diminish by a constant proportion at each stage, the resulting sequence follows a geometric structure. Each new value in the sequence is obtained by applying a fixed multiplier to the preceding term. This regular, proportional decline type is often used to represent processes involving gradual loss, such as energy dissipation or reduction in amplitude over time.When analyzing the total effect of such a process across unlimited iterations, the series of values is referred to as an infinite geometric series. Suppose the magnitude of the common multiplier&#8212;known as the common ratio&#8212;is less than one. In that case, the values in the sequence continue to decrease, and their accumulated total approaches a definite number. This condition ensures convergence rather than unbounded growth or fluctuation.The formula to determine the total sum of an infinite geometric series is:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:7.85%;'><img style='aspect-ratio:289/153;' src='https://cloudfront.jove.com/files/ftp_upload/40095/40095_Equation_1_editor_40095.svg' alt='40095_Equation_1.svg' /></figure>&#8203;where S represents the sum of the series, a is the initial value, and r is the common ratio. This mathematical expression allows for efficient calculation of long-term outcomes in processes that exhibit exponential decrease. Applications span a variety of fields, including physics, economics, and engineering, where such decay models are essential for accurate prediction and analysis.

In systems where values diminish by a constant proportion at each stage, the resulting sequence follows a geometric structure. Each new value in the ...

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Introduction to Sequences and Series

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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.

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

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