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Chapter 11: Problem 26

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$ 6,1,-4,-9, \dots $$

Short Answer

Expert verified
The 20th term of the sequence is -89.

Step by step solution

01

Find the Common Difference

The common difference (d) of an arithmetic sequence is the difference between consecutive terms. It is easy to see in this case that d = -5, since each term subtracts 5 from the previous one.
02

Formulate the nth Term Formula

The formula for the nth term of an arithmetic sequence is given by \[a_{n} = a_{1} + (n-1)d\]. Here, \(a_{1}\) is the first term of the sequence, which is 6, and d is the common difference, which is -5.
03

Find the 20th term

To find the 20th term, substitute \(n = 20\), \(a_{1} = 6\), and \(d = -5\) into the formula: \[a_{20} = 6 + (20-1)(-5)\]

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Key Concepts

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

Common Difference
Understanding the concept of common difference is key to grasping arithmetic sequences. In essence, it's the steady gap between any two successive numbers in the sequence. Specifically, if you subtract a term from the one that follows it, the result is the common difference, denoted as 'd'. For example, given the sequence 6, 1, -4, -9, and so on, we subtract each number from the number that succeeds it (e.g., 1 - 6 = -5 or -4 - 1 = -5), we consistently get -5 as the common difference. It serves as a unique identifier of the sequence's pattern and is vital for further calculations.
Nth Term of an Arithmetic Sequence
The nth term of an arithmetic sequence can be thought of as a way to predict the value of a sequence at any position, n. To find this, you use a direct formula: \(a_n = a_1 + (n-1)d\). Here, \(a_n\) is the nth term you're looking for, \(a_1\) represents the very first term of the sequence and 'd' is the common difference. It's a straightforward approach: you merely take the first term and add the common difference multiplied by one less than the position of the term you want to find. This formula is powerful; it allows you to leap to any term in the sequence without sequentially adding the common difference.
Arithmetic Sequence
An arithmetic sequence is a series of numbers in which each term after the first is obtained by adding a constant, the common difference, to the previous term. It's a pattern of numbers with a distinct, unchanging interval between them. A classic example of an arithmetic sequence is the sequence of even numbers: 2, 4, 6, 8, etc., where each number is obtained by adding 2 to the previous number. The simplicity of arithmetic sequences makes them a fundamental concept in mathematics, as they can be used to model real-world situations and solve various algebraic problems.
Sequence Term Calculation
To practically apply our understanding of arithmetic sequences, we engage in sequence term calculation. This process involves using the nth term formula to find the value of any term in the sequence, given its position, n. For instance, if we want to calculate the 20th term of our example sequence 6, 1, -4, -9, ..., we plug in the first term (6), the common difference (-5), and the position (20): \(a_{20} = 6 + (20-1)(-5)\).After applying simple arithmetic, \(a_{20}\) equates to -89. This illustrates the effectiveness of the nth term formula 鈥� an essential tool in the arsenal of sequence term calculation.

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Most popular questions from this chapter

What is the common ratio in a geometric sequence?

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.

Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=3 \cdot 5^{n} .\) Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}},\) and \(\frac{a_{5}}{a_{4}} .\) What do you observe?

Explaining the Concepts The president of a large company with \(10,000\) employees is considering mandatory cocaine testing for every employee. The test that would be used is \(90 \%\) accurate, meaning that it will detect \(90 \%\) of the cocaine users who are tested, and that \(90 \%\) of the nonusers will test negative. This also means that the test gives \(10 \%\) false positive. Suppose that \(1 \%\) of the employees actually use cocaine. Find the probability that someone who tests positive for cocaine use is, indeed, a user. Hint: Find the following probability fraction: the number of employees who test positive and are cocaine users the number of employees who test positive This fraction is given by $$ 90 \% \text { of } 1 \% \text { of } 10,000 $$ the number who test positive who actually use cocaine plus the number who test positive who do not use cocaine What does this probability indicate in terms of the percentage of employees who test positive who are not actually users? Discuss these numbers in terms of the issue of mandatory drug testing. Write a paper either in favor of or against mandatory drug testing, incorporating the actual percentage accuracy for such tests.

Explaining the Concepts Describe the difference between theoretical probability and empirical probability.
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