Question: Consider the following method that recursively calculates the sum of the elements in an array:

public static int arraySum(int[] arr, int index) {
    if (index == arr.length - 1) {
        return arr[index];
    } else {
        return arr[index] + arraySum(arr, index + 1);
    }
}

Assume that the arr parameter is always a non-null array of integers.

a) Write the base case(s) for this recursive method.

b) Explain why it is necessary to have base cases in recursive methods.

c) Provide an example input and step-by-step execution trace of the arraySum method.

Answer:

a) The base case(s) for this recursive method are:

• When the index parameter is equal to the last valid index of the array (arr.length - 1). In this case, the method returns the value of the last element in the array.
• When the array is empty (arr.length == 0). In this case, the method should return 0, as there are no elements to sum.

b) Base cases are necessary in recursive methods to provide a stopping condition for the recursion. Without base cases, the recursion would continue indefinitely, leading to a stack overflow error in most cases. By defining base cases, we ensure that the recursive calls eventually end, providing a correct result.

c) Let's consider the example input arr = [2, 5, 9, 3]:

1. The arraySum method is initially called with arr and index set to 0.
2. In the recursive call, the method checks if index is equal to arr.length - 1, which is false.
3. It then returns the value of arr[0] (2) plus the result of the recursive call of arraySum(arr, index + 1).
4. The recursive call is made with index set to 1.
5. The method checks if index is equal to arr.length - 1, which is false.
6. It then returns the value of arr[1] (5) plus the result of the recursive call of arraySum(arr, index + 1).
7. The recursive call is made with index set to 2.
8. The method checks if index is equal to arr.length - 1, which is false.
9. It then returns the value of arr[2] (9) plus the result of the recursive call of arraySum(arr, index + 1).
10. The recursive call is made with index set to 3.
11. The method checks if index is equal to arr.length - 1, which is true.
12. It returns the value of arr[3] (3), which is the last element in the array.
13. Backtracking occurs, and each recursive call adds its result to the previous ones.
14. The final sum is obtained: 2 + 5 + 9 + 3 = 19.

Therefore, for the input arr = [2, 5, 9, 3], the arraySum method returns 19.