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Chapter 13: Problem 4

True or False The notation \(a_{5}\) represents the fifth term of a sequence.

Short Answer

Expert verified
True

Step by step solution

01

- Understand Sequence Notation

In a sequence, each term is represented by the notation with a subscript indicating its position. For example, if we have a sequence defined as \(a_1, a_2, a_3,...\), then \(a_n\) represents the nth term.
02

- Identify the Given Notation

In the given problem, the notation \(a_{5}\) is used. Here, the subscript '5' indicates the position of the term in the sequence.
03

- Determine the Meaning of the Notation

Since \(a_{5}\) uses the subscript '5', it represents the fifth term in the sequence, according to the notation rules discussed in Step 1.
04

- Conclude

Based on our understanding, \(a_{5}\) indeed represents the fifth term of the sequence. Therefore, the statement is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

subscript in sequences
In mathematics, sequences are ordered lists of numbers following a particular pattern or rule. Subscripts are used to indicate the positions of terms within these sequences.
When you see a term like \(a_{n}\), the \(n\) is called the subscript. It tells you the position of the term in the sequence. For example, \(a_{1}\) is the first term, \(a_{2}\) is the second term, and so on.
This notation helps in easily identifying terms without confusion. Remember, understanding the subscript is crucial as it shows the location of each term.
nth term
The nth term of a sequence is a general expression for the term located at the n-th position in the sequence.
For example, in the sequence \(1, 3, 5, 7, ...\), the nth term can be expressed as \(a_{n} = 2n - 1\). If you want to know the 5th term (\(a_{5}\)), simply substitute \(n\) with 5: \(a_{5} = 2(5) - 1 = 9\).
Knowing how to find the nth term allows you to determine the value of any term in the sequence directly, without listing all preceding terms.
sequence definition
A sequence is a set of numbers arranged in a specific order following a rule or pattern. Sequences can be finite or infinite and are often represented using notation such as \(a_{1}, a_{2}, a_{3}, ...\).
Each number in the sequence is called a term. The general form of a sequence can be described by its nth term, which provides a formulaic expression for any given position within the sequence.
Understanding sequence definitions, notation, and how to find the nth term is essential for solving problems and analyzing patterns in mathematics.

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Most popular questions from this chapter

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \left\\{8-\frac{3}{4} n\right\\} $$

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 6\left(-\frac{2}{3}\right)^{k-1} $$

Triangular Numbers A triangular number is a term of the sequence $$ u_{1}=1 \quad u_{n+1}=u_{n}+(n+1) $$ List the first seven triangular numbers.

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 3\left(\frac{3}{2}\right)^{k-1} $$

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ 1+5+5^{2}+\cdots+5^{n-1}=\frac{1}{4}\left(5^{n}-1\right) $$
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