Based On The Pattern, Which Statements Are True? Check All That Apply.- The Value Of $a$ Is -6.- The Value Of $b$ Is $\frac{1}{36}$.- As The Exponents Decrease, Each Previous Value Is Divided By 6.- As The Exponents

Understanding the Pattern

The given pattern involves a sequence of numbers with exponents, and we need to identify which statements are true based on this pattern. To do this, we'll analyze the given statements and compare them with the pattern.

The Pattern

The pattern is not explicitly given, but we can infer it from the statements. Let's assume the pattern is:

$$a^b, a^{b-1}, a^{b-2}, \ldots$$

where $a$ and $b$ are constants.

Statement 1: The value of $a$ is -6.

To determine if this statement is true, we need to examine the pattern and see if the value of $a$ is indeed -6.

Let's assume the first term in the pattern is $a^b$. Since the exponents decrease by 1 in each subsequent term, we can write the pattern as:

$$a^b, a^{b-1}, a^{b-2}, \ldots$$

Now, let's consider the second term in the pattern, which is $a^{b-1}$. We can rewrite this as:

$$a^{b-1} = \frac{a^b}{a}$$

Since the pattern involves division by 6 as the exponents decrease, we can set up the following equation:

$$\frac{a^b}{a} = \frac{a^b}{6}$$

Simplifying this equation, we get:

$$a = 6$$

However, this contradicts the statement that the value of $a$ is -6. Therefore, Statement 1 is false.

Statement 2: The value of $b$ is $\frac{1}{36}$.

To determine if this statement is true, we need to examine the pattern and see if the value of $b$ is indeed $\frac{1}{36}$.

Let's assume the first term in the pattern is $a^b$. Since the exponents decrease by 1 in each subsequent term, we can write the pattern as:

$$a^b, a^{b-1}, a^{b-2}, \ldots$$

Now, let's consider the second term in the pattern, which is $a^{b-1}$. We can rewrite this as:

$$a^{b-1} = \frac{a^b}{a}$$

Since the pattern involves division by 6 as the exponents decrease, we can set up the following equation:

$$\frac{a^b}{a} = \frac{a^b}{6}$$

Simplifying this equation, we get:

$$a = 6$$

However, this does not provide any information about the value of $b$. To determine the value of $b$, we need to examine the pattern more closely.

Let's consider the first term in the pattern, which is $a^b$. We can rewrite this as:

$$a^b = a \cdot a \cdot a \cdots (b \text{ times})$$

Since the pattern involves division by 6 as the exponents decrease, we can set up the following equation:

$$a \cdot a \cdot a \cdots (b \text{ times}) = \frac{a \cdot a \cdot a \cdots (b \text{ times})}{6}$$

Simplifying this equation, we get:

$$b = \frac{1}{36}$$

However, this is not a valid solution, as $b$ must be an integer. Therefore, Statement 2 is false.

Statement 3: As the exponents decrease, each previous value is divided by 6.

To determine if this statement is true, we need to examine the pattern and see if each previous value is indeed divided by 6 as the exponents decrease.

Let's assume the first term in the pattern is $a^b$. Since the exponents decrease by 1 in each subsequent term, we can write the pattern as:

$$a^b, a^{b-1}, a^{b-2}, \ldots$$

Now, let's consider the second term in the pattern, which is $a^{b-1}$. We can rewrite this as:

$$a^{b-1} = \frac{a^b}{a}$$

Since the pattern involves division by 6 as the exponents decrease, we can set up the following equation:

$$\frac{a^b}{a} = \frac{a^b}{6}$$

Simplifying this equation, we get:

$$a = 6$$

However, this does not provide any information about the relationship between the terms. To determine the relationship between the terms, we need to examine the pattern more closely.

Let's consider the first term in the pattern, which is $a^b$. We can rewrite this as:

$$a^b = a \cdot a \cdot a \cdots (b \text{ times})$$

Since the pattern involves division by 6 as the exponents decrease, we can set up the following equation:

$$a \cdot a \cdot a \cdots (b \text{ times}) = \frac{a \cdot a \cdot a \cdots (b \text{ times})}{6}$$

Simplifying this equation, we get:

$$\frac{a^b}{a} = \frac{a^b}{6}$$

This shows that each previous value is indeed divided by 6 as the exponents decrease. Therefore, Statement 3 is true.

Conclusion

Q: What is the pattern in the given sequence of numbers?

A: The pattern is not explicitly given, but we can infer it from the statements. Let's assume the pattern is:

$$a^b, a^{b-1}, a^{b-2}, \ldots$$

where $a$ and $b$ are constants.

Q: How do we determine the value of $a$ in the pattern?

A: To determine the value of $a$, we need to examine the pattern and see if the value of $a$ is indeed -6. However, based on the analysis, we found that the value of $a$ is actually 6, not -6. Therefore, the statement that the value of $a$ is -6 is false.

Q: How do we determine the value of $b$ in the pattern?

A: To determine the value of $b$, we need to examine the pattern and see if the value of $b$ is indeed $\frac{1}{36}$. However, based on the analysis, we found that the value of $b$ must be an integer, not a fraction. Therefore, the statement that the value of $b$ is $\frac{1}{36}$ is false.

Q: What is the relationship between the terms in the pattern?

A: The relationship between the terms in the pattern is that each previous value is divided by 6 as the exponents decrease. This can be seen by examining the pattern and setting up the following equation:

$$\frac{a^b}{a} = \frac{a^b}{6}$$

Simplifying this equation, we get:

$$a = 6$$

However, this does not provide any information about the relationship between the terms. To determine the relationship between the terms, we need to examine the pattern more closely.

Let's consider the first term in the pattern, which is $a^b$. We can rewrite this as:

$$a^b = a \cdot a \cdot a \cdots (b \text{ times})$$

Since the pattern involves division by 6 as the exponents decrease, we can set up the following equation:

$$a \cdot a \cdot a \cdots (b \text{ times}) = \frac{a \cdot a \cdot a \cdots (b \text{ times})}{6}$$

Simplifying this equation, we get:

$$\frac{a^b}{a} = \frac{a^b}{6}$$

This shows that each previous value is indeed divided by 6 as the exponents decrease.

Q: What is the significance of the pattern in the given sequence of numbers?

A: The pattern in the given sequence of numbers is significant because it helps us understand the relationship between the terms. By examining the pattern, we can determine the value of $a$ and $b$, and we can also see how each previous value is divided by 6 as the exponents decrease.

Q: How can we apply the pattern to real-world problems?

A: The pattern in the given sequence of numbers can be applied to real-world problems in various fields, such as finance, economics, and engineering. For example, in finance, the pattern can be used to model the growth of investments over time. In economics, the pattern can be used to model the growth of populations over time. In engineering, the pattern can be used to model the growth of systems over time.

Q: What are some common mistakes to avoid when working with patterns?

A: Some common mistakes to avoid when working with patterns include:

• Assuming that the pattern is always linear or exponential
• Failing to examine the pattern closely enough to determine the relationship between the terms
• Making incorrect assumptions about the values of $a$ and $b$
• Failing to consider the significance of the pattern in the given sequence of numbers

By avoiding these common mistakes, we can ensure that we are working with patterns correctly and that we are able to apply the pattern to real-world problems effectively.