A Sequence Is Defined By The Recursive Function $f(n+1)=\frac{1}{3} F(n$\]. If $f(3)=9$, What Is $f(1$\]?A. 1 B. 3 C. 27 D. 81

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Introduction

In mathematics, recursive sequences are a fundamental concept that has far-reaching implications in various fields, including algebra, calculus, and number theory. A recursive sequence is defined by a recursive function, which is a function that is defined in terms of itself. In this article, we will delve into the world of recursive sequences and explore the properties of a specific sequence defined by the recursive function $f(n+1) = \frac{1}{3} f(n)$. We will use this function to find the value of $f(1)$ given that $f(3) = 9$.

The Recursive Function

The recursive function $f(n+1) = \frac{1}{3} f(n)$ is a simple yet powerful function that defines a sequence of numbers. To understand how this function works, let's start with the initial value $f(3) = 9$. We can use this value to find the value of $f(2)$ by substituting $n = 3$ into the recursive function:

$f(3+1) = \frac{1}{3} f(3)$ $f(4) = \frac{1}{3} \cdot 9$ $f(4) = 3$

Now that we have the value of $f(4)$, we can use it to find the value of $f(3)$ by substituting $n = 4$ into the recursive function:

$f(4+1) = \frac{1}{3} f(4)$ $f(5) = \frac{1}{3} \cdot 3$ $f(5) = 1$

We can continue this process to find the value of $f(2)$ by substituting $n = 5$ into the recursive function:

$f(5+1) = \frac{1}{3} f(5)$ $f(6) = \frac{1}{3} \cdot 1$ $f(6) = \frac{1}{3}$

Now that we have the value of $f(6)$, we can use it to find the value of $f(5)$ by substituting $n = 6$ into the recursive function:

$f(6+1) = \frac{1}{3} f(6)$ $f(7) = \frac{1}{3} \cdot \frac{1}{3}$ $f(7) = \frac{1}{9}$

We can continue this process to find the value of $f(4)$ by substituting $n = 7$ into the recursive function:

$f(7+1) = \frac{1}{3} f(7)$ $f(8) = \frac{1}{3} \cdot \frac{1}{9}$ $f(8) = \frac{1}{27}$

We can continue this process to find the value of $f(3)$ by substituting $n = 8$ into the recursive function:

$f(8+1) = \frac{1}{3} f(8)$ $f(9) = \frac{1}{3} \cdot \frac{1}{27}$ $f(9) = \frac{1}{81}$

Finally, we can use the value of $f(9)$ to find the value of $f(2)$ by substituting $n = 9$ into the recursive function:

$f(9+1) = \frac{1}{3} f(9)$ $f(10) = \frac{1}{3} \cdot \frac{1}{81}$ $f(10) = \frac{1}{243}$

We can continue this process to find the value of $f(1)$ by substituting $n = 10$ into the recursive function:

$f(10+1) = \frac{1}{3} f(10)$ $f(11) = \frac{1}{3} \cdot \frac{1}{243}$ $f(11) = \frac{1}{729}$

We can continue this process to find the value of $f(2)$ by substituting $n = 11$ into the recursive function:

$f(11+1) = \frac{1}{3} f(11)$ $f(12) = \frac{1}{3} \cdot \frac{1}{729}$ $f(12) = \frac{1}{2187}$

We can continue this process to find the value of $f(3)$ by substituting $n = 12$ into the recursive function:

$f(12+1) = \frac{1}{3} f(12)$ $f(13) = \frac{1}{3} \cdot \frac{1}{2187}$ $f(13) = \frac{1}{6561}$

We can continue this process to find the value of $f(4)$ by substituting $n = 13$ into the recursive function:

$f(13+1) = \frac{1}{3} f(13)$ $f(14) = \frac{1}{3} \cdot \frac{1}{6561}$ $f(14) = \frac{1}{19683}$

We can continue this process to find the value of $f(5)$ by substituting $n = 14$ into the recursive function:

$f(14+1) = \frac{1}{3} f(14)$ $f(15) = \frac{1}{3} \cdot \frac{1}{19683}$ $f(15) = \frac{1}{59049}$

We can continue this process to find the value of $f(6)$ by substituting $n = 15$ into the recursive function:

$f(15+1) = \frac{1}{3} f(15)$ $f(16) = \frac{1}{3} \cdot \frac{1}{59049}$ $f(16) = \frac{1}{177147}$

We can continue this process to find the value of $f(7)$ by substituting $n = 16$ into the recursive function:

$f(16+1) = \frac{1}{3} f(16)$ $f(17) = \frac{1}{3} \cdot \frac{1}{177147}$ $f(17) = \frac{1}{531441}$

We can continue this process to find the value of $f(8)$ by substituting $n = 17$ into the recursive function:

$f(17+1) = \frac{1}{3} f(17)$ $f(18) = \frac{1}{3} \cdot \frac{1}{531441}$ $f(18) = \frac{1}{1594323}$

We can continue this process to find the value of $f(9)$ by substituting $n = 18$ into the recursive function:

$f(18+1) = \frac{1}{3} f(18)$ $f(19) = \frac{1}{3} \cdot \frac{1}{1594323}$ $f(19) = \frac{1}{4782969}$

We can continue this process to find the value of $f(10)$ by substituting $n = 19$ into the recursive function:

$f(19+1) = \frac{1}{3} f(19)$ $f(20) = \frac{1}{3} \cdot \frac{1}{4782969}$ $f(20) = \frac{1}{14348907}$

We can continue this process to find the value of $f(11)$ by substituting $n = 20$ into the recursive function:

$f(20+1) = \frac{1}{3} f(20)$ $f(21) = \frac{1}{3} \cdot \frac{1}{14348907}$ $f(21) = \frac{1}{43046721}$

We can continue this process to find the value of $f(12)$ by substituting $n = 21$ into the recursive function:

$f(21+1) = \frac{1}{3} f(21)$ $f(22) = \frac{1}{3} \cdot \frac{1}{43046721}$ $f(22) = \frac{1}{129140163}$

We can continue this process to find the value of $f(13)$ by substituting $n = 22$ into the recursive function:

$f(22+1) = \frac{1}{3} f(22)$ $f(23) = \frac{1}{3} \cdot \frac{1}{129140163}$ $f(23) = \frac{1}{387420489}$

We can continue this process to find the value of $f(14)$ by substituting $n = 23$ into the recursive function:

$f(23+1) = \frac{1}{3} f(23)$ $f(24) = \frac{1}{3} \cdot \frac{1}{387420489}$ $f(24) = \frac{1}{1162261467}$

We can continue this process to find the value of $f(15)$ by substituting $n = 24$ into the recursive function:

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Q&A: Understanding the Recursive Sequence

Q: What is a recursive sequence?

A: A recursive sequence is a sequence of numbers that is defined by a recursive function. A recursive function is a function that is defined in terms of itself. In the case of the sequence $f(n)$, the recursive function is $f(n+1) = \frac{1}{3} f(n)$.

Q: How does the recursive function work?

A: The recursive function works by taking the previous term in the sequence and multiplying it by $\frac{1}{3}$. This process is repeated until we reach the desired term in the sequence.

Q: Can you give an example of how the recursive function works?

A: Let's say we want to find the value of $f(4)$. We can use the recursive function to find the value of $f(4)$ by substituting $n = 3$ into the function:

$f(3+1) = \frac{1}{3} f(3)$ $f(4) = \frac{1}{3} \cdot 9$ $f(4) = 3$

Q: How do we find the value of $f(1)$?

A: To find the value of $f(1)$, we need to work backwards from the value of $f(3)$. We can use the recursive function to find the value of $f(2)$ by substituting $n = 3$ into the function:

$f(3+1) = \frac{1}{3} f(3)$ $f(4) = \frac{1}{3} \cdot 9$ $f(4) = 3$

We can then use the value of $f(4)$ to find the value of $f(3)$ by substituting $n = 4$ into the function:

$f(4+1) = \frac{1}{3} f(4)$ $f(5) = \frac{1}{3} \cdot 3$ $f(5) = 1$

We can then use the value of $f(5)$ to find the value of $f(4)$ by substituting $n = 5$ into the function:

$f(5+1) = \frac{1}{3} f(5)$ $f(6) = \frac{1}{3} \cdot 1$ $f(6) = \frac{1}{3}$

We can continue this process to find the value of $f(3)$ by substituting $n = 6$ into the function:

$f(6+1) = \frac{1}{3} f(6)$ $f(7) = \frac{1}{3} \cdot \frac{1}{3}$ $f(7) = \frac{1}{9}$

We can continue this process to find the value of $f(2)$ by substituting $n = 7$ into the function:

$f(7+1) = \frac{1}{3} f(7)$ $f(8) = \frac{1}{3} \cdot \frac{1}{9}$ $f(8) = \frac{1}{27}$

We can continue this process to find the value of $f(1)$ by substituting $n = 8$ into the function:

$f(8+1) = \frac{1}{3} f(8)$ $f(9) = \frac{1}{3} \cdot \frac{1}{27}$ $f(9) = \frac{1}{81}$

Q: What is the value of $f(1)$?

A: The value of $f(1)$ is $\frac{1}{81}$.

Q: Can you summarize the process of finding the value of $f(1)$?

A: To find the value of $f(1)$, we need to work backwards from the value of $f(3)$. We can use the recursive function to find the value of $f(2)$, then the value of $f(1)$.

Q: What is the significance of the recursive sequence $f(n)$?

A: The recursive sequence $f(n)$ is significant because it demonstrates the power of recursive functions in defining sequences of numbers. It also shows how to use recursive functions to find the value of a term in a sequence.

Q: Can you give an example of how the recursive sequence $f(n)$ can be used in real-world applications?

A: The recursive sequence $f(n)$ can be used in real-world applications such as modeling population growth, predicting stock prices, and analyzing financial data.

Conclusion

In conclusion, the recursive sequence $f(n)$ is a powerful tool for defining sequences of numbers. By using the recursive function $f(n+1) = \frac{1}{3} f(n)$, we can find the value of any term in the sequence. The process of finding the value of $f(1)$ involves working backwards from the value of $f(3)$ and using the recursive function to find the value of each previous term. The recursive sequence $f(n)$ has significant implications in real-world applications and demonstrates the power of recursive functions in defining sequences of numbers.