Question

\(\mathrm{A}(\mathrm{n})\) _________ is a special value that marks the end of a series of values.

Short answer

Based on the solution above, give a short answer for the definition and explanation for the term "\(\mathrm{A}(\mathrm{n})\)": The term \(\mathrm{A}(\mathrm{n})\) represents the nth term of a sequence, marking the final term in this series of values. To find \(\mathrm{A}(\mathrm{n})\), we must first identify the type of sequence (arithmetic or geometric) and determine the necessary parameters (first term, common difference, or common ratio). Once these are known, we can use appropriate formulas to calculate the value of the nth term, completing the sequence.

Step-by-step solution

Understanding the given term

\(\mathrm{A}(\mathrm{n})\) denotes the nth term of a sequence. When used in the context of marking the end of a series of values, it suggests that this is the final term in the sequence. The main focus is on understanding the concept of sequences and identifying the final term in a given sequence.

Define sequences and series

A sequence is an ordered list of numbers, often generated by a rule or a pattern. In a sequence, each number is called a term. A series is the sum of all terms in a sequence. In our given problem, we are interested in finding the final term of a sequence, not the sum of the terms.

Notation of a sequence

A common notation for denoting the terms of a sequence is \(\mathrm{A}(\mathrm{n})\), where n represents the position of the term in the sequence (n=1, 2, 3, ...). The value of the nth term can be found by applying a rule or pattern.

Identify the rule or pattern

To find the nth term of a sequence, we first need to identify the rule or pattern generating the terms. This can be, for example, a constant difference between consecutive terms (called an arithmetic progression), a constant ratio between consecutive terms (called a geometric progression), or a more complex pattern.

Finding the nth term

Once the rule or pattern has been identified, use its formula to calculate the value of the nth term, completing the sequence. For example, if the sequence follows an arithmetic progression with the first term \(a_1\) and common difference \(d\), we can find the nth term as: \(\mathrm{A}(n) = a_1 + (n - 1)d\) If the sequence follows a geometric progression with the first term \(a_1\) and common ratio \(r\), we can find the nth term as: \(\mathrm{A}(n) = a_1 \cdot r^{n-1}\)

Applying \(\mathrm{A}(n)\) in the given sequence

Identify or be given the type of sequence, its first term, and either the common difference (arithmetic progression) or common ratio (geometric progression). Use this information to calculate the value of the nth term, or the final term that marks the end of the series of values using formulas from Step 5.

Key concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the 'common difference', to the previous term. This common difference can be positive, negative, or zero, leading to different types of arithmetic progressions.

For example, in the sequence 3, 5, 7, 9, ..., the common difference is 2, since each term is 2 more than the previous term. The general formula to calculate the nth term (denoted as \(A(n)\)) in an arithmetic progression is given by:
\[ A(n) = a_1 + (n - 1)d \]
where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number. This formula helps to find not only the value of any term in the sequence but also the final term that marks the end of a series if the total number of terms is known.

Geometric Progression

A geometric progression (GP), or geometric sequence, is characterized by a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the 'common ratio'. This ratio determines the rate at which the sequence increases or decreases.

Given the sequence 2, 4, 8, 16, ..., the common ratio is 2, because each term is twice the previous term. The formula for the nth term of a geometric progression is:
\[ A(n) = a_1 \cdot r^{(n-1)} \]
Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) indicates the position of the term in the sequence. This allows us to calculate any term within the sequence and to determine the last term in a series when the total number of terms is given.

Series and Sequences Notation

Understanding the notation of sequences is essential for comprehending, communicating, and working with mathematical sequences and series. As demonstrated in the given exercise, the term \(A(n)\) is used to denote the nth term of a sequence, with \(n\) representing the term's position.

The notation carries over to the concept of a series, which is the sum of terms from a sequence. Typically, we write a series using the summation symbol \(\Sigma\), which indicates the addition of a set of numbers. For instance, the sum of the first \(n\) terms of an arithmetic sequence is denoted as \(\Sigma_{i=1}^{n} a_i\), where \(a_i\) represents the sequence's individual terms.

Sequences and series are used not only in pure mathematics but also in various practical contexts such as finance, computer science, and physics, showcasing their far-reaching applications.