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Chapter 9: Problem 9

Find the first four terms and the eighth term of the sequence. $$\left\\{(-1)^{n-1} \frac{n+7}{2 n}\right\\}$$

Short Answer

Expert verified
The first four terms are 4, \(-\frac{9}{4}\), \(\frac{5}{3}\), \(-\frac{11}{8}\), and the eighth term is \(-\frac{15}{16}\).

Step by step solution

01

Understand the Sequence Formula

The sequence given is \((-1)^{n-1} \frac{n+7}{2n}\). This means for each term, you plug in the term number, \(n\), into the formula to get the sequence value.
02

Find the First Term (n=1)

Substitute \(n = 1\) into the formula to calculate the first term: \[(-1)^{1-1} \frac{1+7}{2 \times 1} = 1 \times \frac{8}{2} = 4.\]Thus, the first term is 4.
03

Find the Second Term (n=2)

Substitute \(n = 2\) into the formula to calculate the second term: \[(-1)^{2-1} \frac{2+7}{2 \times 2} = -1 \times \frac{9}{4} = -\frac{9}{4}.\]Thus, the second term is \(-\frac{9}{4}\).
04

Find the Third Term (n=3)

Substitute \(n = 3\) into the formula to calculate the third term: \[(-1)^{3-1} \frac{3+7}{2 \times 3} = 1 \times \frac{10}{6} = \frac{5}{3}.\]Thus, the third term is \(\frac{5}{3}\).
05

Find the Fourth Term (n=4)

Substitute \(n = 4\) into the formula to calculate the fourth term: \[(-1)^{4-1} \frac{4+7}{2 \times 4} = -1 \times \frac{11}{8} = -\frac{11}{8}.\]Thus, the fourth term is \(-\frac{11}{8}\).
06

Find the Eighth Term (n=8)

Substitute \(n = 8\) into the formula to calculate the eighth term: \[(-1)^{8-1} \frac{8+7}{2 \times 8} = -1 \times \frac{15}{16} = -\frac{15}{16}.\]Thus, the eighth term is \(-\frac{15}{16}\).

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Key Concepts

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

Arithmetic Sequences
An arithmetic sequence is a sequence of numbers where each term is obtained by adding a constant difference to the previous term. This constant difference is called the "common difference." If you want to find any term in an arithmetic sequence, you can use the formula: \[ a_n = a_1 + (n-1) imes d \] where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number you wish to find.
  • For example, in the sequence 2, 4, 6, 8..., the common difference is 2.
  • Using the formula, you can calculate the 5th term as 2 + (5-1) 脳 2 = 10.
Unlike the sequence in the exercise, arithmetic sequences do not alternate between positive and negative.
Geometric Sequences
A geometric sequence is defined by each term being multiplied by a fixed number, called the "common ratio," to get the next term. The formula for any term in a geometric sequence can be expressed as: \[ a_n = a_1 imes r^{(n-1)} \] where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
  • For example, in the sequence 3, 6, 12, 24..., the common ratio is 2.
  • Using the formula, the 4th term is 3 脳 2鲁 = 24.
The exercise sequence is not geometric, as seen by the alternating sign pattern and lack of a constant ratio.
Sequence Terms
Sequence terms are individual elements within a sequence, labeled generally as \( a_1, a_2, a_3, \ldots \). Each term in a sequence is determined by its position and the specific formula defining the sequence. In our case, each term is influenced by the formula \((-1)^{n-1} \frac{n+7}{2n}\).
  • The sequence changes each term with alternating signs due to \((-1)^{n-1}\).
  • The term's value partly depends on \(\frac{n+7}{2n}\) which changes with each \(n\).
Plugging in the term number can enable us to calculate the exact value of each sequence term.
Alternating Sequences
An alternating sequence is characterized by its terms flipping sign consistently, often due to factors like \((-1)^n\) or \((-1)^{n-1}\) in the sequence formula. This "alternating" pattern is what makes it different from arithmetic or geometric, where terms grow consistently in one sense or another. In the exercise,
  • The factor \((-1)^{n-1}\) ensures the sequence alternates between positive and negative terms.
  • This is visible when comparing terms like the first term (positive) and the second term (negative).
  • Understanding this alternating characteristic helps predict the sign of any term without computation.
The calculation within the entire sequence will always yield true to this pattern until the end.

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Most popular questions from this chapter

Approximations to \(\pi\) may be obtained from the sequence $$ x_{1}=3, \quad x_{k+1}=x_{k}-\tan x_{k} $$ Use the \((\text { TAN })\) key for tan. (a) Find the first five terms of this sequence. (b) What happens to the terms of the sequence when \(x_{1}=6 ?\)

A pile of logs has 24 logs in the bottom layer, 23 in the second layer, 22 in the third, and so on. The top layer contains 10 logs. Find the total number of logs in the pile.

Find the sum \(S_{n}\) of the arithmetic sequence that satisfies the stated conditions. $$a_{1}=5, \quad a_{20}=9, \quad n=20$$

Graph the sequence. $$\left\\{\frac{1}{n}\right\\}$$

Find the sum \(S_{n}\) of the arithmetic sequence that satisfies the stated conditions. $$a_{1}=-9, \quad a_{10}=15, \quad n=10$$
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