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Chapter 11: Problem 7

Write the first six terms of each arithmetic sequence. $$ a_{1}=\frac{5}{2}, d=-\frac{1}{2} $$

Short Answer

Expert verified
The first six terms of the sequence are \(\frac{5}{2}\), 2, \(\frac{3}{2}\), 1, \(\frac{1}{2}\), 0

Step by step solution

01

Identify first term and common difference

The problem clearly specifies that the first term of the sequence is \(a_{1}=\frac{5}{2}\) and the common difference is \(d=-\frac{1}{2}\). Therefore, no further computation is needed at this point.
02

Apply the formula for the first six terms

The formula for the nth term of an arithmetic sequence is \(a_{n}=a_{1}+(n-1)d\). Substitute the given \(a_{1}\) and \(d\) into the formula for n=1 to n=6 to get the first six terms. The terms will be: \[\begin{align*}a_1 & = a_1 = \frac{5}{2} \a_2 & = a_1 + d = \frac{5}{2} - \frac{1}{2} = 2 \a_3 & = a_2 + d = 2 - \frac{1}{2} = \frac{3}{2} \a_4 & = a_3 + d = \frac{3}{2} - \frac{1}{2} = 1 \a_5 & = a_4 + d = 1 - \frac{1}{2} = \frac{1}{2} \a_6 & = a_5 + d = \frac{1}{2} - \frac{1}{2} = 0\end{align*}\]

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Key Concepts

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

Common Difference
In an arithmetic sequence, the concept of 'common difference' holds great significance. It is the constant difference between consecutive terms in a sequence. If you know the common difference, you can easily predict and calculate any term in the sequence. For instance, if we take an example of the sequence provided in the exercise, the common difference, denoted as \(d\), is \(-\frac{1}{2}\). This value determines how each term varies from the prior term.
  • To find the next term: Add \(d\) to the current term.
  • To check consistency: Measure the difference between terms, ensuring it remains \(d\).

Recognizing and applying the common difference allows you to systematically expand the sequence, either forwards or backwards. This constant is foundational to calculating subsequent sequence terms.
Sequence Terms
Sequence terms are the individual elements or numbers that make up a sequence. In arithmetic sequences, each term follows a specific rule based on its position and the sequence's common difference. Knowing the first term and the common difference, you can generate multiple terms.
  • The first term (\(a_1\)): This is the starting point of the sequence.
  • Subsequent terms: Each is obtained by adding the common difference \(d\) to the previous term.

For example, the initial term here is \(\frac{5}{2}\), and the subsequent sequence terms are obtained by successively adding \(-\frac{1}{2}\). These terms are \(2, \frac{3}{2}, 1, \frac{1}{2},\) and \(0\), demonstrating how each sequence term is interconnected.
Algebraic Formula for Sequences
The algebraic formula for sequences is a powerful tool that helps quickly find terms in an arithmetic sequence without needing to manually add the common difference repeatedly. The formula is given by \(a_n = a_1 + (n-1) \cdot d\), where:
  • \(a_n\): The nth term you want to find.
  • \(a_1\): The first term of the sequence.
  • \(d\): The common difference between the terms.
  • \(n\): The position of the term you're calculating.

By substituting the values into the formula (like in the exercise), you automate the process of finding any number of terms. For this specific situation, the formula helps determine terms eight times faster than relying on repetitive calculations. This simplicity and efficiency make understanding and utilizing the algebraic formula key to effectively navigating arithmetic sequences.

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Most popular questions from this chapter

What is the common ratio in a geometric sequence?

What is an annuity?

Explain how to find the general term of a geometric sequence.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Beginning at 6: 45 A.M., a bus stops on my block every 23 minutes, so I used the formula for the \(n\) th term of an arithmetic sequence to describe the stopping time for the \(n\) th bus of the day.

Use the formula for the value of an annuity to solve Exercises 77鈥�84. Round answers to the nearest dollar. To offer scholarship funds to children of employees, a company invests \(\$ 10,000\) at the end of every three months in an annuity that pays \(10.5 \%\) compounded quarterly. a. How much will the company have in scholarship funds at the end of ten years? b. Find the interest.
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