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Chapter 9: Problem 4

Find the first five terms of the sequence from the formula for \(s_{n}, n \geq 1\) $$(-1)^{n}\left(\frac{1}{2}\right)^{n}$$

Short Answer

Expert verified
The first five terms are: \(-\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \frac{1}{16}, -\frac{1}{32}\).

Step by step solution

01

Understand the formula

The given formula for the sequence is \((-1)^n \left(\frac{1}{2}\right)^n\). This is a sequence where each term alternates in sign and halves in magnitude compared to the previous term.
02

Calculate the first term

Substitute \(n = 1\) into the formula. \[s_{1} = (-1)^{1} \left(\frac{1}{2}\right)^{1} = -\frac{1}{2}\]The first term is \(-\frac{1}{2}\).
03

Calculate the second term

Substitute \(n = 2\) into the formula. \[s_{2} = (-1)^{2} \left(\frac{1}{2}\right)^{2} = \frac{1}{4}\]The second term is \(\frac{1}{4}\).
04

Calculate the third term

Substitute \(n = 3\) into the formula. \[s_{3} = (-1)^{3} \left(\frac{1}{2}\right)^{3} = -\frac{1}{8}\]The third term is \(-\frac{1}{8}\).
05

Calculate the fourth term

Substitute \(n = 4\) into the formula. \[s_{4} = (-1)^{4} \left(\frac{1}{2}\right)^{4} = \frac{1}{16}\]The fourth term is \(\frac{1}{16}\).
06

Calculate the fifth term

Substitute \(n = 5\) into the formula. \[s_{5} = (-1)^{5} \left(\frac{1}{2}\right)^{5} = -\frac{1}{32}\]The fifth term is \(-\frac{1}{32}\).

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Key Concepts

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

Understanding Alternating Sequences
An alternating sequence is a type of sequence where the sign of each term changes with each step. In other words, the sequence alternates between positive and negative values. This creates a back-and-forth pattern in the values of the sequence. In the given sequence formula \((-1)^n\left(\frac{1}{2}\right)^n\), \((-1)^n\) is responsible for the alternating behavior.

Here's how it works:
  • When \(n\) is an odd number, \((-1)^n = -1\), making the term negative.
  • When \(n\) is an even number, \((-1)^n = 1\), making the term positive.
The alternating sign is a key characteristic of such sequences and is crucial for understanding how these sequences behave. Recognizing the alternating pattern assists in easily predicting the nature of subsequent terms without computation.
Geometric Sequence Explained
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the given formula \((-1)^n\left(\frac{1}{2}\right)^n\), the part \(\left(\frac{1}{2}\right)^n\) depicts a geometric sequence
  • where the common ratio is \(\frac{1}{2}\).
This indicates each subsequent term is halved from its predecessor: \(\frac{1}{2} \times \text{previous term}\).
  • For instance, starting from the first positive term \(\frac{1}{4}\), the next positive term will be \(\frac{1}{8}\).
  • The same multiplication logic applies to the negative terms as well.
Understanding the concept of a geometric sequence is vital for recognizing the mathematical pattern and determining how sequences grow or decay over time.
Calculating Terms of a Sequence
Calculating terms in a sequence essentially involves substituting values into the given formula to determine the specific term. For each position \(n\) in the sequence, substitute \(n\) in the formula to find the \(n\)-th term. In the exercise, the sequence formula \((-1)^n\left(\frac{1}{2}\right)^n\) is used to calculate each term
  • by substituting \(n = 1, 2, 3, 4, 5\).
For example:
  • To find the first term: Set \(n = 1\), thus \(s_1 = (-1)^1\left(\frac{1}{2}\right)^1 = -\frac{1}{2}\).
  • Continue this process with \(n = 2, 3, 4, 5\) to obtain the subsequent terms.
By systematically working through the substitutions, one can determine the terms of the sequence accurately. This methodical approach highlights the predictable nature of sequences and emphasizes the importance of formulas in uncovering numerical patterns.

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Most popular questions from this chapter

Explain what is wrong with the statement.The series \(\sum 1 / n^{3 / 2}\) converges by comparison with \(\sum 1 / n^{2}\).

Are true or false. Give an explanation for your answer. \(-5

Give an example of: A monotone sequence that does not converge.

Determine whether the series converges. $$\sum_{n=2}^{\infty} \frac{3}{\ln n^{2}}$$

Decide if the statements are true or false. Give an explanation for your answer. If the sequence \(s_{n}\) of positive terms is unbounded, then the sequence has a term greater than a million.
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