Recursive Sequence: Find F(1) With F(3)=9

Understanding Recursive Functions

Let's dive into the fascinating world of recursive functions and sequences! In mathematics, a recursive function is a function that calls itself in its definition. This might sound a bit mind-boggling at first, but it's a powerful tool for defining sequences where each term depends on the previous one. Our problem involves a sequence defined by the recursive function $f(n+1) = \frac{1}{3} f(n)$. This formula tells us that to get the next term in the sequence, we simply multiply the current term by $\frac{1}{3}$. Think of it like a game of telephone, but with numbers! Each person (or term) gets a value and passes on a modified version to the next. In this case, the modification is always a division by 3. This type of sequence is known as a geometric sequence because there's a constant ratio (in this case, $\frac{1}{3}$) between consecutive terms. The beauty of recursive definitions lies in their ability to express complex patterns with simple, repeating rules. They are fundamental in computer science for algorithms like recursion, and in mathematics for analyzing sequences and series.

Understanding the notation is key here. $f(n+1)$ represents the term that comes after the term $f(n)$. So, if we know the value of $f(n)$, we can directly calculate $f(n+1)$. For example, if $f(1) = 27$, then $f(2) = \frac{1}{3} \times 27 = 9$, and $f(3) = \frac{1}{3} \times 9 = 3$. This step-by-step process is the essence of how recursive sequences unfold. The problem gives us a specific value, $f(3) = 9$, and asks us to find an earlier term, $f(1)$. This means we need to work backward through the sequence.

Working Backwards in a Geometric Sequence

The core of solving this problem lies in our ability to reverse the recursive relationship. The given relation is $f(n+1) = \frac{1}{3} f(n)$. Our goal is to find $f(1)$, and we are given $f(3)$. This means we need to figure out what value of $f(n)$ would result in $f(n+1)$ being 9.

Let's rearrange the formula to solve for $f(n)$:

$f(n+1) = \frac{1}{3} f(n)$

To isolate $f(n)$, we can multiply both sides of the equation by 3:

$3 \times f(n+1) = 3 \times \frac{1}{3} f(n)$

$3 f(n+1) = f(n)$

This rearranged formula is our key to working backward. It tells us that to find the previous term ($f(n)$), we need to multiply the current term ($f(n+1)$) by 3. Now, let's apply this to our problem.

We know $f(3) = 9$. We want to find $f(1)$. First, let's find $f(2)$. Using our rearranged formula, we can set $n+1 = 3$, which means $n = 2$. So, we have:

$f(2) = 3 \times f(3)$

$f(2) = 3 \times 9$

$f(2) = 27$

Excellent! We've found the term that comes before $f(3)$. Now, we need to find the term that comes before $f(2)$, which is $f(1)$. We can use the same rearranged formula. This time, we set $n+1 = 2$, which means $n = 1$. So, we have:

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

$f(1) = 3 \times 27$

$f(1) = 81$

And there you have it! By understanding the recursive definition and how to reverse it, we've successfully determined that $f(1) = 81$. This process highlights the elegance and practicality of working with sequences, allowing us to deduce unknown terms based on a given pattern and a known value. The ability to move both forward and backward in a sequence is a fundamental skill in mathematics and problem-solving.

Verification and Conclusion

To ensure our answer is correct, let's verify our result by working forward from $f(1) = 81$ using the original recursive formula $f(n+1) = \frac{1}{3} f(n)$.

We found $f(1) = 81$.

Now, let's find $f(2)$:

$f(2) = \frac{1}{3} f(1)$

$f(2) = \frac{1}{3} \times 81$

$f(2) = 27$

This matches the value we calculated during our backward steps, which is a good sign!

Next, let's find $f(3)$ using our calculated $f(2)$:

$f(3) = \frac{1}{3} f(2)$

$f(3) = \frac{1}{3} \times 27$

$f(3) = 9$

This also matches the given information in the problem statement ($f(3) = 9$). Therefore, our calculated value for $f(1) = 81$ is indeed correct!

In conclusion, by carefully applying the recursive definition and its inverse, we've confidently solved for the unknown term. This exercise demonstrates the power of mathematical reasoning and the systematic approach required to unravel sequence problems. The relationship $f(n+1) = \frac{1}{3} f(n)$ defines a geometric progression with a common ratio of $\frac{1}{3}$. Given $f(3) = 9$, we used the inverse relationship $f(n) = 3f(n+1)$ to step backward and find $f(2)$ and subsequently $f(1)$. The verification process confirms that our steps were accurate. This problem is a great example of how understanding fundamental mathematical concepts can lead to clear and verifiable solutions.

For further exploration into sequences and series, you can visit resources like Wolfram MathWorld, a comprehensive online encyclopedia of mathematics, or Khan Academy, which offers excellent tutorials on various mathematical topics.