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Chapter 8: Problem 30

In Exercises \(23-34,\) 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. $$a_{1}=-70, d=-5$$

Short Answer

Expert verified
The general term of the given arithmetic sequence is \(a_n = -5n - 65\). The 20th term of the sequence is -165.

Step by step solution

01

Understand and Apply the General Formula

The general formula for the nth term of an arithmetic sequence is given as \(a_n = a_1 + (n-1)*d\), where \(a_1\) is the first term, d is the common difference and n is the term number. Here, \(a_1 = -70\) and \(d = -5\).
02

Substitution for nth term formula.

Substitute the given values into the formula to create the general term of the sequence, to get: \(a_n = -70 + (n-1)*(-5)\). This simplifies to \(a_n = -70 - 5n + 5\). Further simplification gives \(a_n = -5n - 65\). This is the formula for any term of the sequence.
03

Find the 20th term of the sequence

Substitute 20 for n in the formula from Step 2 to find the 20th term. Thus, \(a_{20} = -5*20 - 65 = -100 - 65 = -165\). Therefore, the 20th term of this arithmetic sequence is -165.

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Key Concepts

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

Arithmetic sequence term
Understanding the term of an arithmetic sequence is vital for grasping the overall concept of these sequences. In an arithmetic sequence, each term after the first is created by adding a constant value, called the common difference. For instance, if we start with a number and continually add the same value, this series of numbers is our arithmetic sequence.
To express the nth term of an arithmetic sequence, denoted as \( a_{n} \), we can use a simple formula: \( a_{n} = a_{1} + (n-1) \cdot d \), where \( a_{1} \) is the first term of the sequence, \( d \) is the common difference, and \( n \) signifies the term number. This formula allows us to find any term of the sequence quickly, without calculating all the previous terms.
Sequence general term
The general term of a sequence is essentially a blueprint for generating any term within the sequence by plugging in the desired position number. In the context of the given arithmetic sequence exercise, we determined the general term to be \( a_{n} = -70 - 5(n-1) \). This formula encapsulates the whole sequence by defining how any term can be calculated.
Understanding and deriving the general term is essential in sequence problems as it allows for direct computation of any term. For educational exercises, it's important to pay close attention to simplifying the derived formula, as it makes the use of the formula more straightforward. In the example given, simplifying the general term formula to \( a_{n} = -5n - 65 \) helps students see the pattern clearly鈥攅ach term is obtained by multiplying the term number by -5 and then subtracting 65.
Arithmetic progression
An arithmetic progression is another term for an arithmetic sequence. It's a series of numbers in which the difference of any two successive members is a constant. This characteristic makes arithmetic progressions especially easy to understand and work with in mathematics. For example, the sequence \( -70, -75, -80, -85, ... \) illustrates an arithmetic progression with a common difference of \( d = -5 \).
Arithmetic progressions have a wide range of applications, including in finance for calculating loan repayments, in physics for analyzing uniformly varied motion, and numerous other fields. A solid understanding of arithmetic sequences and their properties, such as sum formulas, is crucial as it forms the foundation for more advanced mathematical concepts.

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Most popular questions from this chapter

The group should select real-world situations where the Fundamental Counting Principle can be applied. These could involve the number of possible student ID numbers on your campus, the number of possible phone numbers in your community, the number of meal options at a local restaurant, the number of ways a person in the group can select outfits for class, the number of ways a condominium can be purchased in a nearby community, and so on. Once situations have been selected, group members should determine in how many ways each part of the task can be done. Group members will need to obtain menus, find out about telephone-digit requirements in the community, count shirts, pants, shoes in closets, visit condominium sales offices, and so on. Once the group reassembles, apply the Fundamental Counting Principle to determine the number of available options in each situation. Because these numbers may be quite large, use a calculator.

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Exercises \(85-87\) will help you prepare for the material covered in the next section. Consider the sequence \(1,-2,4,-8,16, \ldots .\) 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?

Solve by the method of your choice. How many different four-letter passwords can be formed from the letters \(A, B, C, D, E, F,\) and \(G\) if no repetition of letters is allowed?
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