The Formula $f(x+1)=\frac{2}{3}(f(x)$\] Defines A Geometric Sequence Where $f(1)=18$. Which Explicit Formula Can Be Used To Model The Same Sequence?A. $f(x)=(18)\left(\frac{2}{3}\right)(x-1$\] B.

Introduction

In mathematics, a geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The formula $f(x+1)=\frac{2}{3}(f(x))$ defines a geometric sequence, where $f(1)=18$. In this article, we will explore how to find the explicit formula for this sequence.

Understanding the Geometric Sequence Formula

The given formula $f(x+1)=\frac{2}{3}(f(x))$ indicates that each term in the sequence is obtained by multiplying the previous term by $\frac{2}{3}$. This means that the common ratio of the sequence is $\frac{2}{3}$.

Finding the Explicit Formula

To find the explicit formula for the sequence, we need to express the $n^{th}$ term of the sequence in terms of $n$. We can start by using the given formula to find the first few terms of the sequence.

$f(1) = 18$

$f(2) = \frac{2}{3}(f(1)) = \frac{2}{3}(18) = 12$

$f(3) = \frac{2}{3}(f(2)) = \frac{2}{3}(12) = 8$

$f(4) = \frac{2}{3}(f(3)) = \frac{2}{3}(8) = \frac{16}{3}$

Identifying the Pattern

From the first few terms of the sequence, we can observe a pattern. The $n^{th}$ term of the sequence can be expressed as:

$f(n) = 18 \left(\frac{2}{3}\right)^{n-1}$

Verifying the Formula

To verify the formula, we can substitute the values of $n$ into the formula and check if it matches the given sequence.

For $n=1$, $f(1) = 18 \left(\frac{2}{3}\right)^{0} = 18$

For $n=2$, $f(2) = 18 \left(\frac{2}{3}\right)^{1} = 12$

For $n=3$, $f(3) = 18 \left(\frac{2}{3}\right)^{2} = 8$

For $n=4$, $f(4) = 18 \left(\frac{2}{3}\right)^{3} = \frac{16}{3}$

Conclusion

In conclusion, the explicit formula for the geometric sequence defined by the formula $f(x+1)=\frac{2}{3}(f(x))$ is $f(x) = 18 \left(\frac{2}{3}\right)^{x-1}$. This formula can be used to model the same sequence.

Answer

The correct answer is:

A. $f(x) = (18)\left(\frac{2}{3}\right)(x-1)$

However, this is not the correct formula. The correct formula is:

$f(x) = 18 \left(\frac{2}{3}\right)^{x-1}$

Discussion

The formula $f(x+1)=\frac{2}{3}(f(x))$ defines a geometric sequence, where $f(1)=18$. To find the explicit formula for this sequence, we need to express the $n^{th}$ term of the sequence in terms of $n$. We can start by using the given formula to find the first few terms of the sequence. From the first few terms of the sequence, we can observe a pattern. The $n^{th}$ term of the sequence can be expressed as:

$f(n) = 18 \left(\frac{2}{3}\right)^{n-1}$

This formula can be used to model the same sequence.

References

• [1] "Geometric Sequences". Math Open Reference. Retrieved 2023-02-25.
• [2] "Geometric Sequences and Series". Khan Academy. Retrieved 2023-02-25.
  

Introduction

In our previous article, we explored the formula for a geometric sequence and found the explicit formula for the sequence defined by the formula $f(x+1)=\frac{2}{3}(f(x))$, where $f(1)=18$. In this article, we will answer some frequently asked questions about geometric sequences and provide additional information to help you understand this topic better.

Q&A

Q: What is a geometric sequence?

A: A geometric sequence is a type of sequence 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 common ratio of a geometric sequence?

A: The common ratio of a geometric sequence is the number by which each term is multiplied to get the next term. In the formula $f(x+1)=\frac{2}{3}(f(x))$, the common ratio is $\frac{2}{3}$.

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

A: To find the explicit formula for a geometric sequence, you need to express the $n^{th}$ term of the sequence in terms of $n$. You can start by using the given formula to find the first few terms of the sequence and then identify the pattern.

Q: What is the formula for the $n^{th}$ term of a geometric sequence?

A: The formula for the $n^{th}$ term of a geometric sequence is:

$f(n) = a \left(r\right)^{n-1}$

where $a$ is the first term of the sequence and $r$ is the common ratio.

Q: How do I use the formula to find the $n^{th}$ term of a geometric sequence?

A: To use the formula to find the $n^{th}$ term of a geometric sequence, you need to substitute the values of $a$, $r$, and $n$ into the formula.

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

A: An arithmetic sequence is a type of sequence where each term after the first is found by adding a fixed number to the previous term. A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed number.

Q: Can I use the formula for a geometric sequence to model a real-world situation?

A: Yes, you can use the formula for a geometric sequence to model a real-world situation. For example, you can use the formula to model the growth of a population, the decay of a radioactive substance, or the growth of an investment.

Examples

Example 1: Finding the $n^{th}$ term of a geometric sequence

Find the $n^{th}$ term of the geometric sequence defined by the formula $f(x+1)=\frac{2}{3}(f(x))$, where $f(1)=18$.

Solution:

$f(n) = 18 \left(\frac{2}{3}\right)^{n-1}$

Example 2: Using the formula to model a real-world situation

A population of bacteria is growing at a rate of 20% per day. If the initial population is 1000, how many bacteria will there be after 5 days?

Solution:

Let $f(n)$ be the number of bacteria after $n$ days. Then:

$f(n) = 1000 \left(1.2\right)^{n-1}$

Substituting $n=5$, we get:

$f(5) = 1000 \left(1.2\right)^{4} = 1944$

Therefore, there will be 1944 bacteria after 5 days.

Conclusion

In conclusion, geometric sequences are an important topic in mathematics and have many real-world applications. By understanding the formula for a geometric sequence and how to use it, you can model a wide range of situations and make predictions about the future.

References

• [1] "Geometric Sequences". Math Open Reference. Retrieved 2023-02-25.
• [2] "Geometric Sequences and Series". Khan Academy. Retrieved 2023-02-25.