Determining the Function Value f2 Given f6 2/3 and fn n/fn1

Understanding and solving math problems, especially those related to recursive functions, is crucial in various fields, including computer science, mathematics, and even data science. In this article, we will explore how to determine the value of f2 given that f6 2/3 and the function is defined as fn n/fn1. This problem will be solved step-by-step, ensuring clarity and providing insights into the recursive nature of the function.

Understanding the Recursive Function fn n/fn1

The function fn is defined recursively, where each term depends on the previous term. Specifically, it is defined as:

fn n/fn1

To solve for f2, we need to first determine the values of fn for f6, f5, f4, f3, and f2, using the given value of f6 2/3.

Step-by-Step Solution

Step 1: Determine f7

We start with the value of f6, which is given as 2/3. Using the functional equation, we can determine f7 as follows:

f6 6/f7

Rearranging this equation to solve for f7:

f7 6f6/2 6(2/3)/1 9

Step 2: Determine f5

Next, we use f5 5/f6 to find f5:

f5 5 / (2/3) 5 * 3/2 15/2

Step 3: Determine f4

Similarly, we use f4 4/f5 to find f4:

f4 4 / (15/2) 4 * 2/15 8/15

Step 4: Determine f3

We continue to find f3 using f3 3/f4:

f3 3 / (8/15) 3 * 15/8 45/8

Step 5: Determine f2

Finally, we determine f2 using f2 2/f3:

f2 2 / (45/8) 2 * 8/45 16/45

Thus, the value of f2 is 16/45.

Conclusion

By following these steps, we have determined that when f6 2/3 and the recursive function is given by fn n/fn1, the value of f2 is 16/45. Each step demonstrates the recursive nature of the function and the importance of careful calculation in obtaining the correct result.

Additional Resources

For those interested in further exploring such mathematical problems, there are numerous resources available online, including academic papers, textbooks, and interactive online tools that can help deepen understanding.

Remember, solving recursive functions can be both challenging and rewarding, as it involves logical reasoning and precise calculations. With practice, these skills can be greatly improved, making complex problems easier to solve.