Pablo Generates The Function $f(x)=\frac{3}{2}\left(\frac{5}{2}\right)^{x-1}$ To Determine The $x^{\text{th}}$ Number In A Sequence.Which Is An Equivalent Representation?A. $f(x+1)=\frac{5}{2} F(x$\] B. $f(x)=\frac{5}{2}

Introduction

In mathematics, sequences are an essential concept used to describe a list of numbers in a specific order. Pablo has generated a function $f(x)=\frac{3}{2}\left(\frac{5}{2}\right)^{x-1}$ to determine the $x^{\text{th}}$ number in a sequence. However, it is often beneficial to have equivalent representations of a function to facilitate easier calculations and understanding. In this article, we will explore an equivalent representation of Pablo's sequence function.

Understanding Pablo's Sequence Function

Pablo's sequence function is given by $f(x)=\frac{3}{2}\left(\frac{5}{2}\right)^{x-1}$. This function can be used to determine the $x^{\text{th}}$ number in a sequence. To understand how this function works, let's break it down into its components.

• The function has a base of $\frac{5}{2}$, which is raised to the power of $x-1$.
• The function is multiplied by $\frac{3}{2}$, which is a constant factor.

Equivalent Representation of Pablo's Sequence Function

We are given two options for an equivalent representation of Pablo's sequence function:

A. $f(x+1)=\frac{5}{2} f(x)$

B. $f(x)=\frac{5}{2} f(x-1)$

Let's analyze each option to determine which one is an equivalent representation of Pablo's sequence function.

Option A: $f(x+1)=\frac{5}{2} f(x)$

To verify if option A is an equivalent representation, let's substitute $f(x)$ into the equation and simplify.

$f(x+1)=\frac{5}{2} f(x)$

$f(x+1)=\frac{5}{2} \left(\frac{3}{2}\left(\frac{5}{2}\right)^{x-1}\right)$

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

$f(x+1)=\frac{15}{4} \left(\frac{5}{2}\right)^{x-1}$

$f(x+1)=\frac{15}{4} \left(\frac{5}{2}\right)^{x-1} \cdot \frac{2}{2}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

Q&A: Understanding Pablo's Sequence Function

In the previous section, we explored an equivalent representation of Pablo's sequence function. However, we may still have some questions about the function and its equivalent representation. In this section, we will address some of the most frequently asked questions about Pablo's sequence function.

Q: What is Pablo's sequence function?

A: Pablo's sequence function is given by $f(x)=\frac{3}{2}\left(\frac{5}{2}\right)^{x-1}$. This function can be used to determine the $x^{\text{th}}$ number in a sequence.

Q: What is the equivalent representation of Pablo's sequence function?

A: The equivalent representation of Pablo's sequence function is given by $f(x+1)=\frac{5}{2} f(x)$.

Q: How does the equivalent representation work?

A: To understand how the equivalent representation works, let's substitute $f(x)$ into the equation and simplify.

$f(x+1)=\frac{5}{2} f(x)$

$f(x+1)=\frac{5}{2} \left(\frac{3}{2}\left(\frac{5}{2}\right)^{x-1}\right)$

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

$f(x+1)=\frac{15}{4} \left(\frac{5}{2}\right)^{x-1}$

$f(x+1)=\frac{15}{4} \left(\frac{5}{2}\right)^{x-1} \cdot \frac{2}{2}$

$f(x+1)=\frac{15}{8} \left(\frac{5}{2}\right)^{x}$

As we can see, the equivalent representation of Pablo's sequence function is a simple multiplication of the original function by $\frac{5}{2}$.

Q: What are the benefits of having an equivalent representation of Pablo's sequence function?

A: Having an equivalent representation of Pablo's sequence function can have several benefits, including:

• Simplification of calculations: The equivalent representation can make calculations easier and faster.
• Improved understanding: The equivalent representation can provide a deeper understanding of the function and its behavior.
• Flexibility: The equivalent representation can be used in different contexts and applications.

Q: How can I use the equivalent representation of Pablo's sequence function in real-world applications?

A: The equivalent representation of Pablo's sequence function can be used in various real-world applications, including:

• Finance: The equivalent representation can be used to model financial sequences and make predictions about future values.
• Science: The equivalent representation can be used to model scientific sequences and make predictions about future values.
• Engineering: The equivalent representation can be used to model engineering sequences and make predictions about future values.

Conclusion

In conclusion, Pablo's sequence function is a powerful tool for modeling sequences and making predictions about future values. The equivalent representation of the function can be used to simplify calculations, improve understanding, and provide flexibility in different contexts and applications. By understanding the equivalent representation of Pablo's sequence function, we can unlock new possibilities for modeling and predicting sequences in various fields.

Frequently Asked Questions

• Q: What is Pablo's sequence function? A: Pablo's sequence function is given by $f(x)=\frac{3}{2}\left(\frac{5}{2}\right)^{x-1}$.
• Q: What is the equivalent representation of Pablo's sequence function? A: The equivalent representation of Pablo's sequence function is given by $f(x+1)=\frac{5}{2} f(x)$.
• Q: How does the equivalent representation work? A: The equivalent representation works by multiplying the original function by $\frac{5}{2}$.
• Q: What are the benefits of having an equivalent representation of Pablo's sequence function? A: The benefits of having an equivalent representation of Pablo's sequence function include simplification of calculations, improved understanding, and flexibility in different contexts and applications.
• Q: How can I use the equivalent representation of Pablo's sequence function in real-world applications? A: The equivalent representation of Pablo's sequence function can be used in various real-world applications, including finance, science, and engineering.