Which Formula Can Be Used To Describe The Sequence? { -3, \frac{3}{5}, -\frac{3}{25}, \frac{3}{125}, -\frac{3}{625}$}$A. { F(x) = -3\left(\frac{1}{5}\right)^{x-1}$}$B. { F(x) = -3\left(-\frac{1}{5}\right)^{x-1}$}$C.

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Introduction

In mathematics, sequences are an essential concept used to describe a list of numbers in a specific order. Identifying the formula for a given sequence is a crucial skill that helps us understand the underlying pattern and make predictions about future terms. In this article, we will explore a sequence given by βˆ’3,35,βˆ’325,3125,βˆ’3625{-3, \frac{3}{5}, -\frac{3}{25}, \frac{3}{125}, -\frac{3}{625}} and determine which formula can be used to describe it.

Understanding the Sequence

The given sequence consists of five terms, each with a specific pattern. To identify the formula, we need to analyze the sequence and look for any patterns or relationships between the terms.

Upon closer inspection, we can see that each term is obtained by multiplying the previous term by a constant factor. Let's calculate the ratio of consecutive terms to identify this pattern.

  • 35βˆ’3=βˆ’15{\frac{\frac{3}{5}}{-3} = -\frac{1}{5}}
  • βˆ’32535=βˆ’15{\frac{-\frac{3}{25}}{\frac{3}{5}} = -\frac{1}{5}}
  • 3125βˆ’325=βˆ’15{\frac{\frac{3}{125}}{-\frac{3}{25}} = -\frac{1}{5}}
  • βˆ’36253125=βˆ’15{\frac{-\frac{3}{625}}{\frac{3}{125}} = -\frac{1}{5}}

As we can see, the ratio of consecutive terms is always βˆ’15{-\frac{1}{5}}. This indicates that the sequence is a geometric sequence with a common ratio of βˆ’15{-\frac{1}{5}}.

Identifying the Formula

Now that we have identified the common ratio, we can use it to determine the formula for the sequence. A geometric sequence can be described by the formula:

f(x)=aβ‹…rxβˆ’1{f(x) = a \cdot r^{x-1}}

where a{a} is the first term and r{r} is the common ratio.

In this case, the first term is βˆ’3{-3} and the common ratio is βˆ’15{-\frac{1}{5}}. Therefore, the formula for the sequence is:

f(x)=βˆ’3β‹…(βˆ’15)xβˆ’1{f(x) = -3 \cdot \left(-\frac{1}{5}\right)^{x-1}}

Conclusion

In conclusion, the formula that can be used to describe the given sequence is:

f(x)=βˆ’3β‹…(βˆ’15)xβˆ’1{f(x) = -3 \cdot \left(-\frac{1}{5}\right)^{x-1}}

This formula captures the underlying pattern of the sequence and allows us to make predictions about future terms.

Comparison with Other Options

Let's compare our formula with the other options given:

A. f(x)=βˆ’3(15)xβˆ’1{f(x) = -3\left(\frac{1}{5}\right)^{x-1}}

This formula is incorrect because it uses a positive common ratio, whereas our analysis showed that the common ratio is negative.

B. f(x)=βˆ’3(βˆ’15)xβˆ’1{f(x) = -3\left(-\frac{1}{5}\right)^{x-1}}

This formula is correct, but it is not the simplest form. Our formula is more concise and easier to understand.

Final Thoughts

Q: What is a geometric sequence?

A: A geometric sequence is a type of sequence where each term is obtained by multiplying the previous term by a constant factor, known as the common ratio.

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

A: To identify the common ratio, you can calculate the ratio of consecutive terms. If the ratio is constant, then the sequence is a geometric sequence.

Q: What is the formula for a geometric sequence?

A: The formula for a geometric sequence is:

f(x)=aβ‹…rxβˆ’1{f(x) = a \cdot r^{x-1}}

where a{a} is the first term and r{r} is the common ratio.

Q: How do I determine the formula for a given sequence?

A: To determine the formula for a given sequence, you need to identify the first term and the common ratio. Once you have this information, you can use the formula for a geometric sequence to describe the sequence.

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

A: A geometric sequence is a type of sequence where each term is obtained by multiplying the previous term by a constant factor, whereas an arithmetic sequence is a type of sequence where each term is obtained by adding a constant factor to the previous term.

Q: Can a sequence have both a common ratio and a common difference?

A: No, a sequence cannot have both a common ratio and a common difference. If a sequence has a common ratio, it is a geometric sequence, and if it has a common difference, it is an arithmetic sequence.

Q: How do I find the nth term of a geometric sequence?

A: To find the nth term of a geometric sequence, you can use the formula:

f(n)=aβ‹…rnβˆ’1{f(n) = a \cdot r^{n-1}}

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

Q: Can I use the formula for a geometric sequence to describe a sequence that is not geometric?

A: No, the formula for a geometric sequence can only be used to describe a sequence that is geometric. If the sequence is not geometric, you will need to use a different formula or approach to describe it.

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

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

  • Modeling population growth
  • Describing the spread of diseases
  • Analyzing financial data
  • Predicting the behavior of physical systems

Q: Can I use a calculator to find the formula for a geometric sequence?

A: Yes, you can use a calculator to find the formula for a geometric sequence. Many calculators have built-in functions for calculating the nth term of a geometric sequence.

Q: How do I graph a geometric sequence?

A: To graph a geometric sequence, you can use a graphing calculator or a computer program. You can also use a table of values to plot the sequence on a coordinate plane.

Q: Can I use a geometric sequence to model a real-world problem?

A: Yes, you can use a geometric sequence to model a real-world problem. For example, you can use a geometric sequence to model the growth of a population, the spread of a disease, or the behavior of a physical system.