Select The Correct Answer.The Table Below Represents A Geometric Sequence.$\[ \begin{tabular}{|c|c|} \hline $n$ & $f(n)$ \\ \hline 1 & 4 \\ \hline 2 & 20 \\ \hline 3 & 100 \\ \hline \end{tabular} \\]Determine The Recursive Function That

Introduction

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. In this article, we will explore a geometric sequence represented in a table and determine the recursive function that defines it.

Understanding the Geometric Sequence

The table below represents a geometric sequence.

$n$	$f(n)$
1	4
2	20
3	100

In this sequence, each term is obtained by multiplying the previous term by a fixed number. To find the common ratio, we can divide each term by its previous term.

• $f(2) = 20 = f(1) \times r$, where $r$ is the common ratio.
• $f(3) = 100 = f(2) \times r$

Solving for $r$, we get:

• $r = \frac{f(2)}{f(1)} = \frac{20}{4} = 5$
• $r = \frac{f(3)}{f(2)} = \frac{100}{20} = 5$

The common ratio is 5.

Determining the Recursive Function

A recursive function is a function that calls itself to solve a problem. In this case, we want to find the recursive function that defines the geometric sequence.

Let $f(n)$ be the $n$th term of the sequence. We can write the recursive function as:

$f(n) = f(n-1) \times r$

where $r$ is the common ratio.

Substituting the value of $r$, we get:

$f(n) = f(n-1) \times 5$

This is the recursive function that defines the geometric sequence.

Example Use Case

To illustrate the use of the recursive function, let's find the 4th term of the sequence.

$f(4) = f(3) \times 5$ $f(4) = 100 \times 5$ $f(4) = 500$

Therefore, the 4th term of the sequence is 500.

Conclusion

In this article, we explored a geometric sequence represented in a table and determined the recursive function that defines it. We found the common ratio by dividing each term by its previous term and used it to write the recursive function. The recursive function can be used to find any term of the sequence.

Recursive Function Formula

$f(n) = f(n-1) \times 5$

Common Ratio

$r = 5$

Geometric Sequence Table

$n$	$f(n)$
1	4
2	20
3	100

Recursive Function Example

$f(4) = f(3) \times 5$ $f(4) = 100 \times 5$ $f(4) = 500$

Final Answer

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Introduction

In our previous article, we explored a geometric sequence represented in a table and determined the recursive function that defines it. In this article, we will answer some frequently asked questions about geometric sequences and recursive functions.

Q: What is a geometric sequence?

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: How do I find the common ratio of a geometric sequence?

To find the common ratio, you can divide each term by its previous term. For example, if the sequence is 4, 20, 100, you can find the common ratio by dividing 20 by 4, which gives you 5.

Q: What is a recursive function?

A recursive function is a function that calls itself to solve a problem. In the case of a geometric sequence, the recursive function can be used to find any term of the sequence.

Q: How do I write a recursive function for a geometric sequence?

To write a recursive function for a geometric sequence, you need to know the common ratio and the first term of the sequence. The recursive function can be written as:

$f(n) = f(n-1) \times r$

where $r$ is the common ratio.

Q: Can I use a recursive function to find any term of a geometric sequence?

Yes, you can use a recursive function to find any term of a geometric sequence. Simply plug in the value of $n$ into the recursive function and solve for $f(n)$.

Q: What are some examples of geometric sequences?

Some examples of geometric sequences include:

• 2, 6, 18, 54, ...
• 3, 9, 27, 81, ...
• 4, 16, 64, 256, ...

Q: Can I use a geometric sequence to model real-world problems?

Yes, geometric sequences can be used to model real-world problems. For example, if you have a population of bacteria that doubles every hour, you can use a geometric sequence to model the population growth.

Q: What are some common applications of geometric sequences?

Some common applications of geometric sequences include:

• Finance: Geometric sequences can be used to calculate compound interest.
• Biology: Geometric sequences can be used to model population growth.
• Computer Science: Geometric sequences can be used to solve problems in algorithms and data structures.

Q: Can I use a recursive function to solve problems in other areas of mathematics?

Yes, recursive functions can be used to solve problems in other areas of mathematics, such as:

• Algebra: Recursive functions can be used to solve systems of linear equations.
• Calculus: Recursive functions can be used to solve differential equations.
• Number Theory: Recursive functions can be used to solve problems in number theory.

Conclusion

In this article, we answered some frequently asked questions about geometric sequences and recursive functions. We hope that this article has been helpful in understanding these concepts and how they can be applied to real-world problems.

Final Answer

The recursive function that defines a geometric sequence is $f(n) = f(n-1) \times r$, where $r$ is the common ratio.

Common Ratio

$r = 5$

Geometric Sequence Table

$n$	$f(n)$
1	4
2	20
3	100

Recursive Function Example

$f(4) = f(3) \times 5$ $f(4) = 100 \times 5$ $f(4) = 500$