Consider The Sequence: $64, -48, 36, -27, \ldots$Which Formula Can Be Used To Describe The Sequence?A. $f(x+1)=\frac{3}{4} F(x$\]B. $f(x+1)=-\frac{3}{4} F(x$\]C. $f(x)=\frac{3}{4} F(x+1$\]D. $f(x)=-\frac{3}{4}

Introduction

In mathematics, a sequence is a list of numbers in a specific order. Sequences can be described using various formulas, and understanding these formulas is crucial for analyzing and working with sequences. In this article, we will explore a given sequence and determine the formula that can be used to describe it.

The Sequence

The given sequence is: $64, -48, 36, -27, \ldots$ This sequence appears to be obtained by multiplying the previous term by a constant ratio. To determine the formula that describes this sequence, we need to analyze the pattern and identify the common ratio.

Analyzing the Sequence

Let's examine the sequence closely:

• The first term is $64$.
• The second term is $-48$, which is obtained by multiplying the first term by $-\frac{3}{4}$.
• The third term is $36$, which is obtained by multiplying the second term by $-\frac{3}{4}$.
• The fourth term is $-27$, which is obtained by multiplying the third term by $-\frac{3}{4}$.

From this analysis, we can see that each term is obtained by multiplying the previous term by $-\frac{3}{4}$. This indicates that the sequence is a geometric sequence with a common ratio of $-\frac{3}{4}$.

Determining the Formula

A geometric sequence can be described using the formula:

$$f(x) = f(1) \cdot r^{x-1}$$

where $f(1)$ is the first term of the sequence, $r$ is the common ratio, and $x$ is the term number.

In this case, the first term is $64$, and the common ratio is $-\frac{3}{4}$. Therefore, the formula that describes the sequence is:

$$f(x) = 64 \cdot \left(-\frac{3}{4}\right)^{x-1}$$

Conclusion

In conclusion, the formula that can be used to describe the sequence $64, -48, 36, -27, \ldots$ is:

$$f(x) = 64 \cdot \left(-\frac{3}{4}\right)^{x-1}$$

This formula can be used to find any term of the sequence by substituting the desired term number into the formula.

Alternative Formulas

The given options A, B, C, and D are alternative formulas that can be used to describe the sequence. However, these formulas are not as straightforward as the formula we derived.

• Option A: $f(x+1)=\frac{3}{4} f(x)$
• Option B: $f(x+1)=-\frac{3}{4} f(x)$
• Option C: $f(x)=\frac{3}{4} f(x+1)$
• Option D: $f(x)=-\frac{3}{4} f(x+1)$

These formulas can be used to describe the sequence, but they are not as intuitive as the formula we derived.

Discussion

The sequence $64, -48, 36, -27, \ldots$ is a geometric sequence with a common ratio of $-\frac{3}{4}$. The formula that describes this sequence is:

$$f(x) = 64 \cdot \left(-\frac{3}{4}\right)^{x-1}$$

This formula can be used to find any term of the sequence by substituting the desired term number into the formula.

References

• Geometric Sequence
• Mathematics

Related Topics

• Arithmetic Sequence
• Harmonic Sequence
• Mathematical Analysis
  Q&A: Understanding the Sequence and Its Formula =====================================================

Introduction

In our previous article, we explored the sequence $64, -48, 36, -27, \ldots$ and determined the formula that can be used to describe it. In this article, we will answer some frequently asked questions about the sequence and its formula.

Q: What is a geometric sequence?

A: A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: What is the common ratio of the sequence $64, -48, 36, -27, \ldots$?

A: The common ratio of the sequence $64, -48, 36, -27, \ldots$ is $-\frac{3}{4}$.

Q: How do I find the formula for a geometric sequence?

A: To find the formula for a geometric sequence, you need to know the first term and the common ratio. The formula for a geometric sequence is:

$$f(x) = f(1) \cdot r^{x-1}$$

where $f(1)$ is the first term, $r$ is the common ratio, and $x$ is the term number.

Q: How do I use the formula to find a specific term of the sequence?

A: To use the formula to find a specific term of the sequence, you need to substitute the desired term number into the formula. For example, to find the 5th term of the sequence, you would substitute $x=5$ into the formula:

$$f(5) = 64 \cdot \left(-\frac{3}{4}\right)^{5-1}$$

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

A: A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed number called the common difference.

Q: Can I use the formula to find the sum of a geometric sequence?

A: Yes, you can use the formula to find the sum of a geometric sequence. The sum of a geometric sequence is given by the formula:

$$S_n = \frac{a(1-r^n)}{1-r}$$

where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

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

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

• Compound interest: Geometric sequences can be used to calculate the future value of an investment.
• Population growth: Geometric sequences can be used to model population growth.
• Music: Geometric sequences can be used to create musical patterns.

Conclusion

In conclusion, the sequence $64, -48, 36, -27, \ldots$ is a geometric sequence with a common ratio of $-\frac{3}{4}$. The formula that describes this sequence is:

$$f(x) = 64 \cdot \left(-\frac{3}{4}\right)^{x-1}$$

This formula can be used to find any term of the sequence by substituting the desired term number into the formula.

References

• Geometric Sequence
• Arithmetic Sequence
• Mathematics

Related Topics

• Compound Interest
• Population Growth
• Music Theory