Understanding the Geometric Sequence

Oct 13, 2025 by windailys.com 109 views

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.

{ \begin{tabular}{|c|c|} \hline $n$ & $f(n)$ \\ \hline 1 & 4 \\ \hline 2 & 20 \\ \hline 3 & 100 \\ \hline \end{tabular} \}

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.

Finding the Common Ratio

Let's find the common ratio by dividing each term by its previous term.

$20 \div 4 = 5$
$100 \div 20 = 5$

The common ratio is 5.

Recursive Function

A recursive function is a function that calls itself to solve a problem. In this case, we can define a recursive function that takes an integer $n$ as input and returns the $n$ th term of the geometric sequence.

The Recursive Function

Let's define the recursive function $f(n)$ as follows:

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

where $f(n-1)$ is the $(n-1)$ th term of the sequence.

Base Case

The base case is the initial term of the sequence, which is $f(1) = 4$ .

Recursive Formula

The recursive formula is:

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

where $n \geq 2$ .

Example

Let's find the 4th term of the sequence using the recursive function.

$f(4) = f(3) \times 5$ $f(3) = f(2) \times 5$ $f(2) = f(1) \times 5$ $f(1) = 4$

$f(4) = (f(2) \times 5) \times 5$ $f(2) = (f(1) \times 5) \times 5$ $f(1) = 4$

$f(4) = ((f(1) \times 5) \times 5) \times 5$ $f(1) = 4$

$f(4) = (4 \times 5) \times 5 \times 5$ $f(4) = 20 \times 5 \times 5$ $f(4) = 100 \times 5$ $f(4) = 500$

The 4th term of the sequence is 500.

Conclusion

In this article, we have explored a geometric sequence represented in a table and determined the recursive function that defines it. We have found the common ratio of the sequence, defined the recursive function, and used it to find the 4th term of the sequence.

Geometric Sequence and Recursive Function Formula

The recursive function that defines the geometric sequence is:

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

where $n \geq 2$ and $f(1) = 4$ .

The common ratio of the sequence is 5.

Geometric Sequence and Recursive Function Example

Let's find the 4th term of the sequence using the recursive function.

$f(4) = f(3) \times 5$ $f(3) = f(2) \times 5$ $f(2) = f(1) \times 5$ $f(1) = 4$

$f(4) = (f(2) \times 5) \times 5$ $f(2) = (f(1) \times 5) \times 5$ $f(1) = 4$

$f(4) = ((f(1) \times 5) \times 5) \times 5$ $f(1) = 4$

$f(4) = (4 \times 5) \times 5 \times 5$ $f(4) = 20 \times 5 \times 5$ $f(4) = 100 \times 5$ $f(4) = 500$

The 4th term of the sequence is 500.

Geometric Sequence and Recursive Function Code

Here is a Python code that implements the recursive function:

def f(n):
    if n == 1:
        return 4
    else:
        return f(n-1) * 5

print(f(4))

When you run this code, it will print the 4th term of the sequence, which is 500.

Geometric Sequence and Recursive Function Summary

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

where $n \geq 2$ and $f(1) = 4$ .

The common ratio of the sequence is 5.

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 in a geometric sequence?

A: The common ratio is a fixed number that is multiplied by each term to get the next term in the sequence.

Q: How do you find the common ratio in a geometric sequence?

A: To find the common ratio, you can divide each term by its previous term.

Q: What is a recursive function?

A: A recursive function is a function that calls itself to solve a problem.

Q: How do you define a recursive function for a geometric sequence?

A: To define a recursive function for a geometric sequence, you need to specify the base case and the recursive formula.

Q: What is the base case in a recursive function?

A: The base case is the initial term of the sequence, which is used as the starting point for the recursive function.

Q: What is the recursive formula in a recursive function?

A: The recursive formula is the formula that is used to calculate each term in the sequence, based on the previous term.

Q: How do you use a recursive function to find the nth term of a geometric sequence?

A: To use a recursive function to find the nth term of a geometric sequence, you need to specify the base case and the recursive formula, and then use the function to calculate the nth term.

Q: What is the formula for the nth term of a geometric sequence?

A: The formula for the nth term of a geometric sequence is:

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

where $f(n-1)$ is the $(n-1)$ th term of the sequence, and $r$ is the common ratio.

Q: How do you find the common ratio in a geometric sequence if you are given the first two terms?

A: To find the common ratio in a geometric sequence if you are given the first two terms, you can divide the second term by the first term.

Q: What is the relationship between the common ratio and the nth term of a geometric sequence?

A: The common ratio is the factor by which each term is multiplied to get the next term in the sequence.

Q: How do you use the recursive function to find the nth term of a geometric sequence if you are given the first term and the common ratio?

A: To use the recursive function to find the nth term of a geometric sequence if you are given the first term and the common ratio, you need to specify the base case and the recursive formula, and then use the function to calculate the nth term.

Q: What is the formula for the nth term of a geometric sequence if you are given the first term and the common ratio?

A: The formula for the nth term of a geometric sequence if you are given the first term and the common ratio is:

$f(n) = f(1) \times r^{n-1}$

where $f(1)$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: How do you use the formula to find the nth term of a geometric sequence if you are given the first term and the common ratio?

A: To use the formula to find the nth term of a geometric sequence if you are given the first term and the common ratio, you need to plug in the values of $f(1)$ , $r$ , and $n$ into the formula.

Q: What is the relationship between the first term and the common ratio in a geometric sequence?

A: The first term is the starting point of the sequence, and the common ratio is the factor by which each term is multiplied to get the next term.

Q: How do you use the recursive function to find the nth term of a geometric sequence if you are given the first term and the common ratio?

Q: What is the formula for the nth term of a geometric sequence if you are given the first term and the common ratio?

A: The formula for the nth term of a geometric sequence if you are given the first term and the common ratio is:

$f(n) = f(1) \times r^{n-1}$

where $f(1)$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: How do you use the formula to find the nth term of a geometric sequence if you are given the first term and the common ratio?

A: To use the formula to find the nth term of a geometric sequence if you are given the first term and the common ratio, you need to plug in the values of $f(1)$ , $r$ , and $n$ into the formula.

Q: What is the relationship between the first term and the common ratio in a geometric sequence?

A: The first term is the starting point of the sequence, and the common ratio is the factor by which each term is multiplied to get the next term.

Q: How do you use the recursive function to find the nth term of a geometric sequence if you are given the first term and the common ratio?

Q: What is the formula for the nth term of a geometric sequence if you are given the first term and the common ratio?

A: The formula for the nth term of a geometric sequence if you are given the first term and the common ratio is:

$f(n) = f(1) \times r^{n-1}$

where $f(1)$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: How do you use the formula to find the nth term of a geometric sequence if you are given the first term and the common ratio?

A: To use the formula to find the nth term of a geometric sequence if you are given the first term and the common ratio, you need to plug in the values of $f(1)$ , $r$ , and $n$ into the formula.

Q: What is the relationship between the first term and the common ratio in a geometric sequence?

A: The first term is the starting point of the sequence, and the common ratio is the factor by which each term is multiplied to get the next term.

Q: How do you use the recursive function to find the nth term of a geometric sequence if you are given the first term and the common ratio?

Q: What is the formula for the nth term of a geometric sequence if you are given the first term and the common ratio?

A: The formula for the nth term of a geometric sequence if you are given the first term and the common ratio is:

$f(n) = f(1) \times r^{n-1}$

where $f(1)$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: How do you use the formula to find the nth term of a geometric sequence if you are given the first term and the common ratio?

A: To use the formula to find the nth term of a geometric sequence if you are given the first term and the common ratio, you need to plug in the values of $f(1)$ , $r$ , and $n$ into the formula.

Q: What is the relationship between the first term and the common ratio in a geometric sequence?

A: The first term is the starting point of the sequence, and the common ratio is the factor by which each term is multiplied to get the next term.

Q: How do you use the recursive function to find the nth term of a geometric sequence if you are given the first term and the common ratio?

Q: What is the formula for the nth term of a geometric sequence if you are given the first term and the common ratio?

A: The formula for the nth term of a geometric sequence if you are given the first term and the common ratio is:

$f(n) = f(1) \times r^{n-1}$

where $f(1)$ is the first term, $r$ is the common ratio, and $n$ is the term number.