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Chapter 11: Problem 2

write the first four terms of each sequence whose general term is given. $$ a_{n}=4 n-1 $$

Short Answer

Expert verified
The first four terms of the sequence are: 3, 7, 11, 15

Step by step solution

01

Determining the First Term

To find the first term of the sequence i.e. \(a_1\), substitute \(n=1\) in the general term \(a_{n}=4n-1\). So, \(a_1 = 4(1) - 1 = 4 - 1 = 3\)
02

Determining the Second Term

To find the second term of the sequence i.e. \(a_2\), substitute \(n=2\) in the general term \(a_{n}=4n-1\). So, \(a_2 = 4(2) - 1= 8 - 1 = 7\)
03

Determining the Third Term

To find the third term of the sequence i.e. \(a_3\), substitute \(n=3\) in the general term \(a_{n}=4n-1\). So, \(a_3 = 4(3) - 1 = 12 - 1 = 11\)
04

Determining the Fourth Term

To find the fourth term of the sequence i.e. \(a_4\), substitute \(n=4\) in the general term \(a_{n}=4n-1\). So, \(a_4 = 4(4) - 1 = 16 - 1 = 15\)

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Key Concepts

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

Arithmetic Sequence
An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This difference is referred to as the common difference, and it is quite essential as it defines the arithmetic nature of the sequence.

For example, if we start with the number 2 and keep adding 3 to get the next term, the sequence would be 2, 5, 8, 11, and so on. Here, the common difference is 3. Recognizing the pattern of an arithmetic sequence helps us predict subsequent values and the behaviors of the sequence over time.
General Term of a Sequence
The general term of a sequence, often denoted by the variable term 'a_n', provides a formulaic representation allowing you to find any term in the sequence without listing all the preceding terms. This is particularly useful for arithmetic sequences where each term can be calculated independently of the others.

A general term for an arithmetic sequence can be written as: \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference. Calculating the general term is like having a map, letting you jump to any point in the sequence directly.
Finding Terms of a Sequence
To find specific terms of a sequence, we apply the general term formula by substituting the position number we are interested in for \( n \.\)

For instance, if we wish to find the 100th term of an arithmetic sequence without having to write out the first 99 terms, we can simply plug in 100 for \( n \) in the general term formula. This process saves time and allows for quick calculations of terms in long sequences. Always double-check your substitution to ensure the correct term is calculated.
Algebraic Expressions
Algebraic expressions are a fundamental concept in algebra, consisting of numbers, variables (like \( n \) in sequences), and operations (like addition and multiplication). In the context of sequences, these expressions define the rule or pattern by which the sequence is created.

An algebraic expression for a sequence, like the one in the provided example \( a_n = 4n - 1 \), makes it straightforward to compute terms. The formula itself encapsulates the relationship between the terms of the sequence and their position, which is invaluable when you're dealing with sequences that extend to many terms.

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Most popular questions from this chapter

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 \(\$ 10,000\) by correctly matching four numbers drawn from white balls ( 1 through 56 ) and matching the number on the gold Mega Bali (1 through 46 ). What is the probability of winning this consolation prize?

The probability that South Florida will be hit by a major hurricane (category 4 or 5 ) in any single year is \(\frac{1}{16}\) (Source: National Hurricane Center) a. What is the probability that South Florida will be hit by a major hurricane two years in a row? b. What is the probability that South Florida will be hit by a major hurricane in three consecutive years? c. What is the probability that South Florida will not be hit by a major hurricane in the next ten years? d. What is the probability that South Florida will be hit by a major hurricane at least once in the next ten years?

Among all pairs of numbers whose sum is \(24,\) find a pair whose product is as large as possible. What is the maximum product? (Section 3.1, Example 6)

In Exercises \(105-108\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If the \(n\) th term of a geometric sequence is \(a_{n}=3(0.5)^{n-1}\) the common ratio is \(\frac{1}{2}\)

Explain how to find the sum of the first \(n\) terms of a geometric sequence without having to add up all the terms.
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