A Sequence Has A Common Ratio Of 3 2 \frac{3}{2} 2 3 ​ And F ( 5 ) = 81 F(5) = 81 F ( 5 ) = 81 . Which Explicit Formula Represents The Sequence?A. F ( X ) = 24 ( 3 2 ) X − 1 F(x) = 24\left(\frac{3}{2}\right)^{x-1} F ( X ) = 24 ( 2 3 ​ ) X − 1 B. F ( X ) = 16 ( 3 2 ) X − 2 F(x) = 16\left(\frac{3}{2}\right)^{x-2} F ( X ) = 16 ( 2 3 ​ ) X − 2 C. $f(x)

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Understanding the Problem

When dealing with sequences, it's essential to understand the concept of a common ratio and how it affects the terms of the sequence. In this problem, we're given a sequence with a common ratio of $\frac{3}{2}$ and the value of the fifth term, $f(5) = 81$. Our goal is to find the explicit formula that represents this sequence.

The Formula for a Geometric Sequence

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. The general formula for a geometric sequence is:

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

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

Finding the First Term

We're given that the common ratio is $\frac{3}{2}$ and the value of the fifth term is $81$. We can use this information to find the first term, $a$. Plugging in the values we know, we get:

$$f(5) = a \cdot \left(\frac{3}{2}\right)^{5-1} = 81$$

Simplifying the equation, we get:

$$a \cdot \left(\frac{3}{2}\right)^{4} = 81$$

Solving for the First Term

To solve for the first term, $a$, we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by $\left(\frac{3}{2}\right)^{4}$:

$$a = \frac{81}{\left(\frac{3}{2}\right)^{4}}$$

Simplifying the expression, we get:

$$a = \frac{81}{\frac{81}{16}} = 16$$

Finding the Explicit Formula

Now that we have the first term, $a$, we can plug it into the general formula for a geometric sequence:

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

Substituting the values we know, we get:

$$f(x) = 16 \cdot \left(\frac{3}{2}\right)^{x-1}$$

Conclusion

The explicit formula that represents the sequence is:

$$f(x) = 16 \cdot \left(\frac{3}{2}\right)^{x-1}$$

This formula represents the sequence with a common ratio of $\frac{3}{2}$ and the value of the fifth term, $f(5) = 81$.

Comparison with Other Options

Let's compare our answer with the other options:

• Option A: $f(x) = 24\left(\frac{3}{2}\right)^{x-1}$
• Option B: $f(x) = 16\left(\frac{3}{2}\right)^{x-2}$
• Option C: $f(x) = 32\left(\frac{3}{2}\right)^{x-1}$

Our answer, $f(x) = 16 \cdot \left(\frac{3}{2}\right)^{x-1}$, matches option A.

Final Answer

The explicit formula that represents the sequence is:

$$f(x) = 16 \cdot \left(\frac{3}{2}\right)^{x-1}$$

This formula represents the sequence with a common ratio of $\frac{3}{2}$ and the value of the fifth term, $f(5) = 81$.

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Understanding the Problem

When dealing with sequences, it's essential to understand the concept of a common ratio and how it affects the terms of the sequence. In this problem, we're given a sequence with a common ratio of $\frac{3}{2}$ and the value of the fifth term, $f(5) = 81$. Our goal is to find the explicit formula that represents this sequence.

Q&A

Q: What is a geometric sequence?

A: 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.

Q: What is the general formula for a geometric sequence?

A: The general formula for a geometric sequence is:

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

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

Q: How do we find the first term, $a$, in a geometric sequence?

A: To find the first term, $a$, we can use the formula:

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

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

Q: What is the common ratio, $r$, in this problem?

A: The common ratio, $r$, is $\frac{3}{2}$.

Q: What is the value of the fifth term, $f(5)$, in this problem?

A: The value of the fifth term, $f(5)$, is $81$.

Q: How do we find the explicit formula for the sequence?

A: To find the explicit formula for the sequence, we can plug the values we know into the general formula for a geometric sequence:

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

Substituting the values we know, we get:

$$f(x) = 16 \cdot \left(\frac{3}{2}\right)^{x-1}$$

Q: What is the explicit formula for the sequence?

A: The explicit formula for the sequence is:

$$f(x) = 16 \cdot \left(\frac{3}{2}\right)^{x-1}$$

Q: How do we compare the explicit formula with other options?

A: To compare the explicit formula with other options, we can plug the values we know into the other options and see if they match the explicit formula.

Q: What are the other options?

A: The other options are:

• Option A: $f(x) = 24\left(\frac{3}{2}\right)^{x-1}$
• Option B: $f(x) = 16\left(\frac{3}{2}\right)^{x-2}$
• Option C: $f(x) = 32\left(\frac{3}{2}\right)^{x-1}$

Q: Which option matches the explicit formula?

A: Option A matches the explicit formula.

Conclusion

The explicit formula that represents the sequence is:

$$f(x) = 16 \cdot \left(\frac{3}{2}\right)^{x-1}$$

This formula represents the sequence with a common ratio of $\frac{3}{2}$ and the value of the fifth term, $f(5) = 81$.