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Chapter 9: Problem 11

Determine whether the sequence is geometric. If so, then find the common ratio. $$1,-\sqrt{7}, 7,-7 \sqrt{7}, \dots$$

Short Answer

Expert verified
Yes, the given sequence is geometric and the common ratio is -\(\sqrt{7}\)

Step by step solution

01

Identifying Sequence Type

Compare the ratio of successive terms in the sequence. Use the formula Ratio = \( \frac{2nd term}{1st term}\) = \( \frac{3rd term}{2nd term}\) = \( \frac{4th term}{3rd term}\) and so on.
02

Calculate Ratio

By applying the formula, Ratio = \(\frac{-\sqrt{7}}{1}\) = \(\frac{7}{-\sqrt{7}}\) = \(\frac{-7\sqrt{7}}{7}\). These all simplify to -\(\sqrt{7}\).
03

Conclusion

Since all ratios are equal, the sequence is geometric and the common ratio is -\(\sqrt{7}\).

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Key Concepts

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

Understanding the Common Ratio
In geometric sequences, the common ratio is a critical concept. It's the factor by which we multiply one term to get to the next term. Identifying this ratio helps us determine if a sequence is geometric. The formula for finding the common ratio is: \( \text{common ratio} = \frac{\text{next term}}{\text{previous term}} \).

For example, let's look at a sequence: 1, \(-\sqrt{7}\), 7, \(-7\sqrt{7}\), and so on. We calculate the common ratio from each pair of consecutive terms:
  • First pair: \( \frac{-\sqrt{7}}{1} = -\sqrt{7}\)
  • Second pair: \( \frac{7}{-\sqrt{7}} = -\sqrt{7}\)
  • Third pair: \( \frac{-7\sqrt{7}}{7} = -\sqrt{7}\)
Notice how each ratio simplifies to \(-\sqrt{7}\)? This confirms the sequence is geometric because each term maintains the same ratio.
The Concept of a Sequence
A sequence is an ordered list of numbers. These numbers are called terms, and they can follow various patterns or rules.

Geometric sequences specifically amplify the idea of a constant multiplication factor, or common ratio, being applied. Understanding sequences involves recognizing the type based on how each term relates to its neighbors. For instance:
  • In an arithmetic sequence, each term is developed by adding or subtracting a fixed number from the previous term.
  • In a geometric sequence, each term results from multiplying the previous term by a fixed non-zero number, known as the common ratio.
Recognizing these patterns allows us to see relationships that may not be immediately apparent, aiding in further analysis or problem-solving.
Math Problem-Solving with Geometric Sequences
Solving problems in mathematics often involves identifying patterns or relationships within data. Geometric sequences provide a clear example of this. Recognizing that a sequence is geometric can simplify many math problems, as knowing the common ratio allows for predictions and computations.

Here's how to tackle a geometric sequence math problem:
  • First, confirm the sequence type by checking if consecutive terms have a constant ratio.
  • Then, compute the common ratio using the formula \( \frac{\text{next term}}{\text{previous term}} \).
  • Finally, use the common ratio to address the problem, whether it's finding future terms, solving for a particular term, or even summing parts of the sequence.
These steps provide a framework for efficiently addressing math problems involving sequences, stripping down complex problems to manageable pieces.

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Most popular questions from this chapter

Use the following definition of the arithmetic mean \(\bar{x}\) of a set of \(n\) measurements \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) \(\bar{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}\) Find the arithmetic mean of the six checking account balances 327.15 dollar ,785.69 dollar, 433.04 dollar, 265.38 dollar, 604.12 dollar and 590.30 dollar. Use the statistical capabilities of a graphing utility to verify your result.

Finding the Probability of a Complement You are given the probability that an event will happen. Find the probability that the event will not happen. $$P(E)=\frac{2}{3}$$

Consider \(n\) independent trials of an experiment in which each trial has two possible outcomes: "success" or "failure." The probability of a success on each trial is \(p,\) and the probability of a failure is \(q=1-p .\) In this context, the term \(_{n} C_{k} p^{k} q^{n-k}\) in the expansion of \((p+q)^{n}\) gives the probability of \(k\) successes in the \(n\) trials of the experiment.The probability of a sales representative making a sale with any one customer is \(\frac{1}{3} .\) The sales representative makes eight contacts a day. To find the probability of making four sales, evaluate the term $$_{8} C_{4}\left(\frac{1}{3}\right)^{4}\left(\frac{2}{3}\right)^{4}$$, in the expansion of \(\left(\frac{1}{3}+\frac{2}{3}\right)^{8}\).

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True or False? Determine whether the statement is true or false. Justify your answer. If \(A\) and \(B\) are independent events with nonzero probabilities, then \(A\) can occur when \(B\) occurs.
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