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

Sequences are ordered lists of numbers arranged according to a specific rule or pattern. The nth term is found using a formula based on its position.For example, the height of a bouncing ball decreases with each bounce, forming a decreasing sequence in which each height is a fixed fraction of the previous one.Each number in a sequence is called a term, and the position of the ordered terms determines its value.When the pattern is clear, dots indicate the sequence continues.Some sequences define terms using previous ones, called recursive sequences. For example, the nth term is defined using the (n-1)th term.For instance, in the Fibonacci sequence, each term equals the sum of the two preceding terms.Partial sums are the sums of the first few terms of a sequence. Often shown using sigma notation, partial sums help analyze how the sum grows as more terms are added.Each of these is called a partial sum: S1 is the first, S2 is the second, and Sn is the nth&#8217;s term summation.The sequence formed by these is called the sequence of partial sums.For example, each week's deposit is a term when tracking weekly savings. Partial sums show how total savings grow over time.

sequences

Sequences

Sequences are fundamental mathematical objects consisting of ordered lists of numbers that follow a specific rule or pattern. Sequences are critical in various mathematical concepts, including calculus, series, and number theory. They can model real-world phenomena such as population growth, financial investments, and physical processes like the diminishing height of a bouncing ball.Each number in a sequence is referred to as a term. Typically, the terms are denoted as a1, a2, a3,&#8230;, where the subscript indicates the position within the sequence. When the pattern is evident, sequences often include dots to signify continuation.Mathematically, a sequence can also be seen as a function whose domain is the set of natural numbers, with each natural number mapped to a specific term. This functional representation is useful when defining sequences explicitly or recursively.Recursive sequences define each term based on previous terms. One of the most famous examples is the Fibonacci sequence, where each term equals the sum of the two preceding terms:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:35.42%;'><img style='aspect-ratio:1244/72;' src='https://cloudfront.jove.com/files/ftp_upload/40093/40093_Equation_1_editor_40093.svg' alt='40093_Equation_1.svg' /></figure>This definition emphasizes the dependence of each term on its predecessors, which is characteristic of recursive sequences.Partial sums are the sums of the first few terms of a sequence and are valuable for analyzing how the cumulative sum evolves as more terms are added. For a sequence {an}, the nth partial sum is given by:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:33.86%;'><img style='aspect-ratio:1109/132;' src='https://cloudfront.jove.com/files/ftp_upload/40093/40093_Equation_2_editor_40093.svg' alt='40093_Equation_2.svg' /></figure>Understanding sequences, recursive definitions, and partial sums forms the basis for exploring more advanced topics such as infinite series, convergence, and mathematical induction.Further, a telescoping sequence is a special type of sequence in which most terms cancel out when the partial sum is expanded, making it easier to evaluate the sum. In a telescoping series, the nth partial sum often reduces to the difference between a few terms, typically the first and the last, allowing for a simplified closed-form expression. This property is especially useful in evaluating infinite series and proving convergence.

Sequences are fundamental mathematical objects consisting of ordered lists of numbers that follow a specific rule or pattern. Sequences are critical in ...

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JoVE Core

Math Fundamentals

Unit

Introduction to Sequences and Series

KEY TERMS AND CONCEPTS

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 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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