A Sequence Is Defined By The Recursive Formula $f(n+1) = 1.5 F(n$\]. Which Sequence Could Be Generated Using The Formula?A. $-12, -18, -27, \ldots$B. $-20, 30, -45, \ldots$C. $-18, -16.5, -1.5, \ldots$D. $-16,

In mathematics, a recursive sequence is a sequence that is defined by a recursive formula, which is a formula that defines each term of the sequence as a function of the preceding terms. The recursive formula for a sequence is often denoted as $f(n+1) = a \cdot f(n)$, where $a$ is a constant and $f(n)$ is the $n$th term of the sequence.

The Given Recursive Formula

The given recursive formula is $f(n+1) = 1.5 f(n)$. This formula indicates that each term of the sequence is obtained by multiplying the previous term by 1.5.

Analyzing the Options

Let's analyze the options given to determine which sequence could be generated using the formula.

Option A: $-12, -18, -27, \ldots$

To determine if this sequence could be generated using the formula, we need to check if each term is obtained by multiplying the previous term by 1.5.

• $-18 = 1.5 \cdot -12$
• $-27 = 1.5 \cdot -18$

Since the second term is obtained by multiplying the first term by 1.5, this sequence could be generated using the formula.

Option B: $-20, 30, -45, \ldots$

To determine if this sequence could be generated using the formula, we need to check if each term is obtained by multiplying the previous term by 1.5.

• $30 = 1.5 \cdot -20$ (This is incorrect, as 30 is not equal to 1.5 times -20)
• $-45 = 1.5 \cdot 30$ (This is incorrect, as -45 is not equal to 1.5 times 30)

Since the second term is not obtained by multiplying the first term by 1.5, this sequence could not be generated using the formula.

Option C: $-18, -16.5, -1.5, \ldots$

To determine if this sequence could be generated using the formula, we need to check if each term is obtained by multiplying the previous term by 1.5.

• $-16.5 = 1.5 \cdot -18$ (This is incorrect, as -16.5 is not equal to 1.5 times -18)
• $-1.5 = 1.5 \cdot -16.5$ (This is incorrect, as -1.5 is not equal to 1.5 times -16.5)

Since the second term is not obtained by multiplying the first term by 1.5, this sequence could not be generated using the formula.

Option D: $-16, -24, -36, \ldots$

To determine if this sequence could be generated using the formula, we need to check if each term is obtained by multiplying the previous term by 1.5.

• $-24 = 1.5 \cdot -16$
• $-36 = 1.5 \cdot -24$

Since the second term is obtained by multiplying the first term by 1.5, this sequence could be generated using the formula.

Conclusion

Based on the analysis of the options, the sequence that could be generated using the formula $f(n+1) = 1.5 f(n)$ is:

• $-12, -18, -27, \ldots$

This sequence is obtained by multiplying each term by 1.5 to obtain the next term.

Understanding the Formula

The formula $f(n+1) = 1.5 f(n)$ indicates that each term of the sequence is obtained by multiplying the previous term by 1.5. This means that the sequence will grow exponentially, with each term being 1.5 times the previous term.

Example Use Case

The formula $f(n+1) = 1.5 f(n)$ can be used to model a variety of real-world phenomena, such as population growth or financial investments. For example, if we have a population that grows by 50% each year, we can use the formula to model the population growth over time.

Conclusion

In conclusion, the sequence that could be generated using the formula $f(n+1) = 1.5 f(n)$ is $-12, -18, -27, \ldots$. This sequence is obtained by multiplying each term by 1.5 to obtain the next term. The formula can be used to model a variety of real-world phenomena, such as population growth or financial investments.