Provide a recursive definition of some sequence of numbers. …

Provide a recursive definition of some sequence of numbers. Choose one different from that of any posted thus far. Write a recursive method that given n, computes the nth term of that sequence. Also provide an equivalent iterative implementation. How do the two implementations compare?

Answer

In order to provide a recursive definition of a sequence of numbers different from the ones posted so far, let’s consider the Fibonacci sequence. The Fibonacci sequence is a well-known sequence of numbers in which each number is the sum of the two preceding ones, starting with either 0 and 1 or 1 and 1.

Here is the recursive definition of the Fibonacci sequence:

1. Base case 1: F(i) = 0, if i = 0.
2. Base case 2: F(i) = 1, if i = 1.
3. Recursive case: F(i) = F(i-1) + F(i-2), if i > 1.

Now, let’s proceed to implement a recursive method that computes the nth term of the Fibonacci sequence.

“`
public static int fibonacciRecursive(int n) {
if (n == 0) {
return 0;
} else if (n == 1) {
return 1;
} else {
return fibonacciRecursive(n – 1) + fibonacciRecursive(n – 2);
}
}
“`

In this implementation, the base cases correspond to the first two terms of the Fibonacci sequence (0 and 1) and the recursive case calculates the nth term by recursively calling the method with n-1 and n-2.

Alternatively, we can implement an equivalent iterative version of the Fibonacci sequence.

“`
public static int fibonacciIterative(int n) {
if (n == 0) {
return 0;
} else if (n == 1) {
return 1;
} else {
int previous = 0;
int current = 1;
int result = 0;

for (int i = 2; i <= n; i++) { result = previous + current; previous = current; current = result; } return result; } } ``` In this iterative implementation, we initialize the previous and current variables to represent the first two terms of the Fibonacci sequence (0 and 1). We then use a for loop to calculate the nth term iteratively by updating the variables based on the Fibonacci formula. Comparing the two implementations, we can observe the following: 1. Recursive method: The recursive implementation follows the recursive definition of the Fibonacci sequence closely. However, it may not be efficient for large values of n due to repeated function calls and redundant calculations. 2. Iterative method: The iterative implementation avoids the exponential time complexity of the recursive version by calculating the terms one by one, using the Fibonacci formula. It is generally more efficient and recommended for practical use.