Problem 87 Write each set using set-builder... [FREE SOLUTION] (2024)

Chapter 1: Problem 87

Write each set using set-builder notation. Answers may vary. $$\\{6,8,10,12, \dots, 82\\}$$

Short Answer

Expert verified

\[ \{ 4 + 2n \mid n \in \mathbb{N}, 1 \leq n \leq 39 \} \]

Step by step solution

Identify the Pattern

Determine the pattern of the set. Notice that the elements are increasing by 2 in each consecutive term. The set starts at 6 and ends at 82.

Define the Element in Terms of n

Express the general term of the set. Since the terms are increasing by 2, we can write the general term as: \[ a_n = 6 + 2(n-1) \]where $ n $ is a positive integer.

Simplify the General Term

Simplify the expression for $ a_n $:\[ a_n = 4 + 2n \]As $ n $ begins from 1, this describes all elements in the set.

Define the Range for n

Determine the limits for $ n $. The smallest element is 6, and the largest is 82. Solve for $ n $ when $ a_n = 82 $:\[ 82 = 4 + 2n \]\[ 78 = 2n \]\[ n = 39 \]Thus, $ n $ ranges from 1 to 39.

Write in Set-Builder Notation

Using the simplified general term and the range for $ n $, write the set in set-builder notation:\[ \{ 6, 8, 10, 12, \dots, 82 \} = \{ 4 + 2n \mid n \in \mathbb{N}, 1 \leq n \leq 39 \} \]

Key Concepts

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

Arithmetical Sequences

An arithmetical sequence is a list of numbers in which the difference between consecutive terms is constant. For instance, in the sequence $ \{6, 8, 10, 12, \, \dots, 82\} $, each term after the first is obtained by adding 2 to the previous term. This constant difference is known as the *common difference*.
The formula for finding any term in an arithmetical sequence is:
\[ a_n = a_1 + (n-1)d \]
where:

$ a_n $ represents the nth term
$ a_1 $ is the first term
$ d $ is the common difference

In our example, the first term $ a_1 $ is 6, and the common difference $ d $ is 2. So, the general term (or nth term) can be written as $ a_n = 6 + 2(n-1) $. By simplifying this formula, we get $ a_n = 4 + 2n $.
Understanding arithmetical sequences is crucial because it helps us describe patterns and make predictions about the series.

General Term of a Sequence

The general term of a sequence, sometimes referred to as the nth term, is an expression that allows you to find any term in the sequence without listing all previous terms. For arithmetical sequences, we use a specific formula.
For the sequence $ \{6, 8, 10, 12, \, \dots, 82\} $, the general term was derived and simplified to $ a_n = 4 + 2n $. This formula indicates that no matter which term you're looking for, as long as you know its position $ n $, you can substitute it into the formula to find the term's value.
To put this into practice, if we wanted to find the 10th term of our sequence, we would substitute $ n = 10 $ into our general term formula:
\[ a_{10} = 4 + 2(10) = 4 + 20 = 24 \]
This means the 10th term is 24. Knowing the general term simplifies the process of identifying any term in a given sequence.

Natural Numbers

Natural numbers are the set of positive integers beginning from 1 and continuing indefinitely: $ \{1, 2, 3, 4, \, \dots\} $. They are fundamental in number theory and are used to count objects.
When defining elements of a set in set-builder notation, the range for $ n $ is often given in terms of natural numbers to specify valid positions. In our example, the general term for the sequence $ \{6, 8, 10, 12, \, \dots, 82\} $ uses the range $ 1 \leq n \leq 39 $, where $ n \in \mathbb{N} $. This means that $ n $ is a natural number, and it must be between 1 and 39 inclusive.
Natural numbers are easy to work with and play a key role in many mathematical contexts, ensuring clarity and precision in definitions and operations.

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