Name: arithmetic sequence formulas and partial sums
Uploaded: 2025-11-26T05:43:53.000Z
Duration: 1 min 30 s
Channel: JoVE

An arithmetic sequence is a list of numbers where each term increases or decreases by the same fixed number, known as the common difference. Consider a pile of poles. The first layer contains 25 poles, and the number of poles continues to decrease by 1 in each successive layer.Given that the pile has 12 layers, the goal is to find the total number of poles.This arrangement forms an arithmetic sequence, as the number of poles decreases by a constant amount from one layer to the next.In this scenario, the number of poles in the 12th layer is calculated using the formula for the nth term of an arithmetic sequence, based on the first term, the common difference, and the number of layers. The values of these terms are then substituted into the formula, which simplifies to 25 minus 11, giving 14 poles in the 12th layer.The total number of poles in the pile, known as the partial sum of the sequence, is then calculated by taking the average of the number of poles in the first and last layers and multiplying it by the total number of layers. It is termed a partial sum since only the first 12 terms of the sequence are added. This results in a partial sum of 12 multiplied by the average of 25 and 14, yielding 234 poles.

arithmetic sequences

Arithmetic Sequences

An arithmetic sequence is a structured arrangement of numbers where each term is derived by adding a constant value, known as the common difference, to the previous term. This consistent pattern allows for the efficient computation of any term within the sequence as well as the cumulative sum of multiple terms. The formula for finding the nth term of an arithmetic sequence is:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:82.68%;'><img style='aspect-ratio:1920/74;' src='https://cloudfront.jove.com/files/ftp_upload/40094/40094_Equation_1_editor_40094.svg' alt='40094_Equation_1.svg' /></figure>Here, a&#8345; represents the nth term of the sequence, a is the first term, d is the common difference, and n is the term number or position in the sequence. This equation is essential for determining the value of any term without listing all preceding terms. To compute the total of the first n terms, known as the partial sum, either of the following formulas is used:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/163;' src='https://cloudfront.jove.com/files/ftp_upload/40094/40094_Equation_2_editor_40094.svg' alt='40094_Equation_2.svg' /></figure>In these expressions, S&#8345; denotes the sum of the first n terms, and a&#8345; again refers to the nth term, derived using the earlier formula. These formulas offer a concise and systematic approach for analyzing regularly spaced numerical patterns in both theoretical and practical applications.

An arithmetic sequence is a structured arrangement of numbers where each term is derived by adding a constant value, known as the common difference, to ...

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Math Fundamentals

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Introduction to Sequences and Series

KEY TERMS AND CONCEPTS

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.

The total of the first n terms is called the partial sum. To find it, you can use either of two formulas. In these expressions, S n means the sum of the first n terms, and a n is the nth term found from the earlier formula. These tools give a clear way to analyze evenly spaced numerical patterns in math and in practical situations.

Arithmetic Sequence Formulas and Partial Sums

arithmetic-sequence-formulas-and-partial-sums

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 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 arithmetic sequences use a common difference, the nth-term formula, and partial sum formulas to find terms and totals in a number pattern.

Arithmetic Sequence Formulas 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

Arithmetic Sequence Formulas and Partial Sums

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