Continuing the Backward Calculation

Oct 13, 2025 by windailys.com 86 views

A Recursive Sequence: Unraveling the Mystery of f(n)

In mathematics, recursive sequences are a fundamental concept that helps us understand the behavior of functions and their values at different points. 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 explore a specific recursive sequence defined by the 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 $$f(n+1)=\frac{1}{3} f(n)$$ is a simple yet powerful function that defines the sequence. To understand how this function works, let's break it down step by step.

The function takes an input value $$n$$ and returns a value $$f(n+1)$$.
The value of $$f(n+1)$$ is calculated by multiplying the value of $$f(n)$$ by $$\frac{1}{3}$$.
This means that each subsequent value in the sequence is one-third of the previous value.

Now that we understand the recursive function, let's use it to find the value of $$f(1)$$. We are given that $$f(3)=9$$, and we want to find $$f(1)$$. To do this, we need to work our way backwards from $$f(3)$$ to $$f(1)$$.

We know that $$f(3)=9$$, so we can use this value to find $$f(2)$$.
Since $$f(n+1)=\frac{1}{3} f(n)$$, we can write $$f(3)=\frac{1}{3} f(2)$$.
Solving for $$f(2)$$, we get $$f(2)=3f(3)=3(9)=27$$.

Continuing the Backward Calculation

Now that we have found $$f(2)=27$$, we can use this value to find $$f(1)$$.

We know that $$f(2)=27$$, so we can use this value to find $$f(1)$$.
Since $$f(n+1)=\frac{1}{3} f(n)$$, we can write $$f(2)=\frac{1}{3} f(1)$$.
Solving for $$f(1)$$, we get $$f(1)=3f(2)=3(27)=81$$.

In this article, we explored a recursive sequence defined by the function $$f(n+1)=\frac{1}{3} f(n)$$. We used this function to find the value of $$f(1)$$, given that $$f(3)=9$$. By working our way backwards from $$f(3)$$ to $$f(1)$$, we found that $$f(1)=81$$. This example illustrates the power of recursive sequences and the importance of understanding how they work.

The final answer is: $\boxed{81}$
A Recursive Sequence: Unraveling the Mystery of f(n) - Q&A

In our previous article, we explored a recursive sequence defined by the function $$f(n+1)=\frac{1}{3} f(n)$$. We used this function to find the value of $$f(1)$$, given that $$f(3)=9$$. In this article, we will answer some frequently asked questions about recursive sequences and the function $$f(n+1)=\frac{1}{3} f(n)$$.

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.

Q: How does the function $$f(n+1)=\frac{1}{3} f(n)$$ work?

A: The function $$f(n+1)=\frac{1}{3} f(n)$$ takes an input value $$n$$ and returns a value $$f(n+1)$$. The value of $$f(n+1)$$ is calculated by multiplying the value of $$f(n)$$ by $$\frac{1}{3}$$.

Q: Can you explain the concept of recursion in more detail?

A: Recursion is a fundamental concept in mathematics that helps us understand how functions work. A recursive function is a function that is defined in terms of itself. This means that the function is defined in a way that it calls itself repeatedly until it reaches a base case.

Q: What is the base case in the function $$f(n+1)=\frac{1}{3} f(n)$$?

A: The base case in the function $$f(n+1)=\frac{1}{3} f(n)$$ is the value of $$f(1)$$. This is because the function is defined in terms of $$f(n+1)$$, and we need to know the value of $$f(1)$$ to calculate the value of $$f(2)$$.

Q: How do you find the value of $$f(1)$$ using the function $$f(n+1)=\frac{1}{3} f(n)$$?

A: To find the value of $$f(1)$$ using the function $$f(n+1)=\frac{1}{3} f(n)$$, we need to work our way backwards from $$f(3)$$ to $$f(1)$$. We know that $$f(3)=9$$, so we can use this value to find $$f(2)$$, and then use $$f(2)$$ to find $$f(1)$$.

Q: Can you provide an example of how to find the value of $$f(1)$$ using the function $$f(n+1)=\frac{1}{3} f(n)$$?

A: Yes, here is an example of how to find the value of $$f(1)$$ using the function $$f(n+1)=\frac{1}{3} f(n)$$:

We know that $$f(3)=9$$, so we can use this value to find $$f(2)$$.
Since $$f(n+1)=\frac{1}{3} f(n)$$, we can write $$f(3)=\frac{1}{3} f(2)$$.
Solving for $$f(2)$$, we get $$f(2)=3f(3)=3(9)=27$$.
Now that we have found $$f(2)=27$$, we can use this value to find $$f(1)$$.
Since $$f(n+1)=\frac{1}{3} f(n)$$, we can write $$f(2)=\frac{1}{3} f(1)$$.
Solving for $$f(1)$$, we get $$f(1)=3f(2)=3(27)=81$$.

In this article, we answered some frequently asked questions about recursive sequences and the function $$f(n+1)=\frac{1}{3} f(n)$$. We explained the concept of recursion, the base case, and how to find the value of $$f(1)$$ using the function $$f(n+1)=\frac{1}{3} f(n)$$. We hope that this article has helped you understand recursive sequences and the function $$f(n+1)=\frac{1}{3} f(n)$$ better.

The final answer is: $\boxed{81}$