Select The Correct Answer.Which Sequence Is Geometric And Has $\frac{1}{2}$ As The Common Ratio?A. $\quad \ldots, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$B. $\quad \ldots, \frac{1}{4}, \frac{1}{2}, 1, 2,

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Introduction

In mathematics, 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. This concept is crucial in various mathematical operations, including algebra, calculus, and statistics. In this article, we will explore geometric sequences and common ratios, focusing on identifying the correct sequence with a common ratio of $\frac{1}{2}$.

What is a Geometric Sequence?

A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number, known as the common ratio. The general form of a geometric sequence is:

a,ar,ar2,ar3,…a, ar, ar^2, ar^3, \ldots

where $a$ is the first term, and $r$ is the common ratio.

Common Ratios

The common ratio is a crucial component of a geometric sequence. It determines how each term is obtained from the previous term. In a geometric sequence, the common ratio can be:

  • A positive number, indicating that each term is obtained by multiplying the previous term by a positive number.
  • A negative number, indicating that each term is obtained by multiplying the previous term by a negative number.
  • A fraction, indicating that each term is obtained by multiplying the previous term by a fraction.

Identifying Geometric Sequences

To identify a geometric sequence, we need to examine the relationship between consecutive terms. If the ratio between consecutive terms is constant, then the sequence is geometric.

Example 1: A Geometric Sequence with a Common Ratio of $\frac{1}{2}$

Let's consider the sequence:

…,1,12,14,18,…\ldots, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots

To determine if this sequence is geometric, we need to examine the ratio between consecutive terms.

121=12\frac{\frac{1}{2}}{1} = \frac{1}{2}

1412=12\frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}

1814=12\frac{\frac{1}{8}}{\frac{1}{4}} = \frac{1}{2}

As we can see, the ratio between consecutive terms is constant, which means that this sequence is geometric.

Example 2: A Geometric Sequence with a Common Ratio of $\frac{1}{2}$

Let's consider the sequence:

…,14,12,1,2,…\ldots, \frac{1}{4}, \frac{1}{2}, 1, 2, \ldots

To determine if this sequence is geometric, we need to examine the ratio between consecutive terms.

1214=2\frac{\frac{1}{2}}{\frac{1}{4}} = 2

112=2\frac{1}{\frac{1}{2}} = 2

21=2\frac{2}{1} = 2

As we can see, the ratio between consecutive terms is not constant, which means that this sequence is not geometric.

Conclusion

In conclusion, a geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number, known as the common ratio. The common ratio can be a positive number, a negative number, or a fraction. To identify a geometric sequence, we need to examine the ratio between consecutive terms. If the ratio is constant, then the sequence is geometric. In this article, we explored geometric sequences and common ratios, focusing on identifying the correct sequence with a common ratio of $\frac{1}{2}$.

Final Answer

Based on the analysis above, the correct sequence with a common ratio of $\frac{1}{2}$ is:

…,1,12,14,18,…\ldots, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots

This sequence is geometric because the ratio between consecutive terms is constant, which is $\frac{1}{2}$.

Discussion

Introduction

In our previous article, we explored geometric sequences and common ratios, focusing on identifying the correct sequence with a common ratio of $\frac{1}{2}$. In this article, we will answer some frequently asked questions about geometric sequences and common ratios.

Q&A

Q: What is the difference between a geometric sequence and an arithmetic sequence?

A: A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number, known as the common ratio. An arithmetic sequence, on the other hand, is a sequence of numbers where each term is obtained by adding a fixed number, known as the common difference.

Q: How do I identify a geometric sequence?

A: To identify a geometric sequence, you need to examine the ratio between consecutive terms. If the ratio is constant, then the sequence is geometric.

Q: What is the formula for a geometric sequence?

A: The general form of a geometric sequence is:

a,ar,ar2,ar3,…a, ar, ar^2, ar^3, \ldots

where $a$ is the first term, and $r$ is the common ratio.

Q: What is the formula for the nth term of a geometric sequence?

A: The formula for the nth term of a geometric sequence is:

an=a⋅rn−1a_n = a \cdot r^{n-1}

where $a$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: How do I find the common ratio of a geometric sequence?

A: To find the common ratio of a geometric sequence, you need to examine the ratio between consecutive terms. If the ratio is constant, then you can use the formula:

r=an+1anr = \frac{a_{n+1}}{a_n}

where $a_n$ is the nth term, and $a_{n+1}$ is the (n+1)th term.

Q: What are some real-world applications of geometric sequences and common ratios?

A: Geometric sequences and common ratios have many real-world applications, including:

  • Finance: Compound interest and investment returns
  • Biology: Population growth and decay
  • Physics: Sound waves and vibrations
  • Computer Science: Algorithm design and analysis

Q: How do I use geometric sequences and common ratios in real-world problems?

A: To use geometric sequences and common ratios in real-world problems, you need to:

  • Identify the problem and the relevant variables
  • Determine the type of sequence (geometric or arithmetic)
  • Use the formula for the nth term or the common ratio to solve the problem
  • Interpret the results and draw conclusions

Conclusion

In conclusion, geometric sequences and common ratios are fundamental concepts in mathematics that have many real-world applications. By understanding these concepts, you can solve problems in finance, biology, physics, and computer science. We hope that this Q&A article has helped you to better understand geometric sequences and common ratios.

Final Tips

  • Practice, practice, practice: The more you practice, the more comfortable you will become with geometric sequences and common ratios.
  • Use real-world examples: Try to relate geometric sequences and common ratios to real-world problems and applications.
  • Ask questions: Don't be afraid to ask questions if you are unsure about a concept or a problem.

Discussion

What are some other real-world applications of geometric sequences and common ratios? How do you use geometric sequences and common ratios in your daily life? Share your thoughts and examples in the comments below!