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Chapter 10: Problem 1

Write the first four terms of each sequence whose general term is given. $$a_{n}=3 n+2$$

Short Answer

Expert verified
The first four terms of the sequence are 5, 8, 11, 14.

Step by step solution

01

Understanding the general term

The general term of a sequence, also known as the nth term, describes the pattern of the sequence. In this case, the general term is \(a_{n}=3n+2\). This means, each term in the sequence is equal to three times its position in the sequence plus two.
02

Calculate the first term

To calculate the first term (n=1), we substitute n=1 into the formula: \(a_{1}=3*1+2=5\). So, the first term of the sequence is 5.
03

Calculate the second term

Following the same method, we substitute n=2 into the formula: \(a_{2}=3*2+2=8\). So, the second term of the sequence is 8.
04

Calculate the third term

Respectively, we can find the third term by substituting n=3: \(a_{3}=3*3+2=11\). The third term of the sequence is 11.
05

Calculate the fourth term

Finally, we can calculate the fourth term by substituting n=4: \(a_{4}=3*4+2=14\). The fourth term of the sequence is 14.

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Key Concepts

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

General Term
The concept of a general term, often denoted as \(a_n\), serves as the formula that governs the pattern of a sequence. It provides a method to find any specific term within the sequence without listing out all previous terms. By understanding the general term, you gain insight into how the sequence is constructed and can predict any term's value by merely substituting the desired position number into the formula.

In the example \(a_{n} = 3n + 2\), the general term indicates a mathematical rule where each term is calculated by multiplying the position number \(n\) by 3 and then adding 2. This formula makes it clear that each term directly depends on its position in the sequence, which is the essence of the general term's utility.
Nth Term
The nth term of a sequence is a specific term that you can find using the general term. It reflects any term at position \(n\), thereby helping you to compute or discover the value of that term without listing all preceding numbers.

For example, if you are looking for the nth term in a sequence governed by \(a_{n} = 3n + 2\), you can plug in any positive integer into \(n\) to directly find its corresponding term. This is useful when dealing with large sequences, as it saves time and effort by avoiding the need to calculate each preceding term individually.
Arithmetic Sequence
An arithmetic sequence is a type of sequence where each term after the first is produced by adding a constant difference to the previous term. This ensures a linear growth or decline across the sequence.

In the context of the sequence defined by \(a_{n} = 3n + 2\), it is part of an arithmetic sequence because the difference between any two consecutive terms is constant. Calculating the differences for the first four terms expressed by this sequence: 5, 8, 11, 14, you find that the difference is 3, demonstrating a common difference property of arithmetic sequences.

Recognizing an arithmetic sequence is useful in mathematics because it enables straightforward calculations of future terms and helps in understanding linear relationships.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Rather than performing the addition, I used the formula \(S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right)\) to find the sum of the first 30 terms of the sequence \(2,4,8,16,32, \ldots\)

Use mathematical induction to prove that each statement is true for every positive integer \(n\). 2 is a factor of \(n^{2}+3 n\)

Mega Millions is a multi-state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to \(46 .\) Use this information to solve Exercises \(27-30 .\) Express all probabilities as fractions. A player wins a minimum award of \(\$ 150\) by correctly matching three numbers drawn from white balls (1 through 56) and matching the number on the gold Mega Ball" ( 1 through 46 ). What is the probability of winning this consolation prize?

Graph \(y=3 \tan \frac{x}{2}\) for \(-\pi

Use this information to solve Exercises \(47-48 .\) The mathematics department of a college has 8 male professors, 11 female professors, 14 male teaching assistants, and 7 female teaching assistants. If a person is selected at random from the group, find the probability that the selected person is a professor or a female.
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