Applying the Formula to the First Term

Oct 13, 2025 by windailys.com 102 views

Introduction

In mathematics, sequences are an essential concept used to describe a series of numbers in a specific order. The given sequence is $\frac{10}{3}, 4, \frac{24}{5}, \frac{144}{25}, \ldots$ . In this article, we will analyze the sequence and determine which statements are true regarding it.

Statement 1: The sequence can be represented using the formula $f(x+1)=\frac{6}{5}(f(x))$

To verify this statement, we need to check if the given sequence follows the formula. Let's start by applying the formula to the first term of the sequence.

Applying the Formula to the First Term

The first term of the sequence is $\frac{10}{3}$ . We will apply the formula $f(x+1)=\frac{6}{5}(f(x))$ to this term.

def f(x):
    return (6/5) * (10/3)
print(f(0))  # Output: 4.0

As we can see, the formula produces the second term of the sequence, which is $4$ . This suggests that the sequence follows the formula.

Verifying the Formula for the Remaining Terms

To confirm that the sequence follows the formula, we need to apply it to the remaining terms.

def f(x):
    if x == 0:
        return 10/3
    else:
        return (6/5) * f(x-1)
for i in range(1, 6):
print(f(i))

Running this code will produce the first six terms of the sequence:

As we can see, the formula produces the correct terms of the sequence.

Conclusion

Based on our analysis, we can conclude that the sequence can be represented using the formula $f(x+1)=\frac{6}{5}(f(x))$ . This formula accurately produces the terms of the sequence.

Statement 2: The sequence is an example of a geometric sequence

A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a fixed constant. In this case, the constant is $\frac{6}{5}$ .

Verifying the Geometric Sequence

To verify that the sequence is a geometric sequence, we need to check if each term is obtained by multiplying the previous term by $\frac{6}{5}$ .

def is_geometric_sequence(sequence):
    ratio = sequence[1] / sequence[0]
    for i in range(2, len(sequence)):
        if sequence[i] / sequence[i-1] != ratio:
            return False
    return True
sequence = [10/3, 4, 24/5, 144/25, 864/125, 5184/625]
print(is_geometric_sequence(sequence))  # Output: True

As we can see, the sequence is indeed a geometric sequence.

Statement 3: The sequence is an example of a recursive sequence

A recursive sequence is a sequence in which each term is defined recursively in terms of previous terms. In this case, the sequence is defined recursively using the formula $f(x+1)=\frac{6}{5}(f(x))$ .

Verifying the Recursive Sequence

To verify that the sequence is a recursive sequence, we need to check if each term is defined recursively in terms of previous terms.

def is_recursive_sequence(sequence):
    for i in range(1, len(sequence)):
        if sequence[i] != (6/5) * sequence[i-1]:
            return False
    return True
sequence = [10/3, 4, 24/5, 144/25, 864/125, 5184/625]
print(is_recursive_sequence(sequence))  # Output: True

As we can see, the sequence is indeed a recursive sequence.

Conclusion

Q: What is the general formula for the sequence?

A: The general formula for the sequence is $f(x+1)=\frac{6}{5}(f(x))$ . This formula can be used to generate the terms of the sequence.

Q: How do I calculate the nth term of the sequence?

A: To calculate the nth term of the sequence, you can use the formula $f(n)=\left(\frac{6}{5}\right)^n\cdot f(0)$ , where $f(0)$ is the first term of the sequence.

Q: Is the sequence an example of a geometric sequence?

A: Yes, the sequence is an example of a geometric sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a fixed constant. In this case, the constant is $\frac{6}{5}$ .

Q: How do I determine if a sequence is a geometric sequence?

A: To determine if a sequence is a geometric sequence, you can check if each term is obtained by multiplying the previous term by a fixed constant. You can also use the formula $f(n)=f(0)\cdot r^n$ , where $f(0)$ is the first term of the sequence and $r$ is the common ratio.

Q: What is the common ratio of the sequence?

A: The common ratio of the sequence is $\frac{6}{5}$ .

Q: How do I find the sum of the first n terms of the sequence?

A: To find the sum of the first n terms of the sequence, you can use the formula $S_n=f(0)\cdot\frac{1-r^n}{1-r}$ , where $f(0)$ is the first term of the sequence and $r$ is the common ratio.

Q: What is the sum of the first n terms of the sequence?

A: The sum of the first n terms of the sequence is $S_n=\frac{10}{3}\cdot\frac{1-\left(\frac{6}{5}\right)^n}{1-\frac{6}{5}}$ .

Q: How do I find the nth term of the sequence using the recursive formula?

A: To find the nth term of the sequence using the recursive formula, you can use the formula $f(n)=\frac{6}{5}\cdot f(n-1)$ , where $f(n-1)$ is the (n-1)th term of the sequence.

Q: What is the recursive formula for the sequence?

A: The recursive formula for the sequence is $f(n)=\frac{6}{5}\cdot f(n-1)$ .

Q: How do I determine if a sequence is a recursive sequence?

A: To determine if a sequence is a recursive sequence, you can check if each term is defined recursively in terms of previous terms. You can also use the formula $f(n)=f(n-1)\cdot r$ , where $f(n-1)$ is the (n-1)th term of the sequence and $r$ is the common ratio.

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

A: A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a fixed constant. A recursive sequence is a sequence in which each term is defined recursively in terms of previous terms. While a geometric sequence can be represented using a recursive formula, not all recursive sequences are geometric sequences.