Question

Find the error(s) in the following recursive method, and explain how to correct it (them). This method should find the sum of the values from 0 to n. 1 public int sum( int n ) 2 { 3 if ( n == 0 ) 4 return 0; 5 else 6 return n + sum( n ); 7 } // end method sum

Short answer

Correct the recursive call to `return n + sum(n - 1)` to prevent infinite recursion.

Step-by-step solution

Identify the Base Case

In this recursive method, the base case has been written as \( n == 0 \), which should correctly return 0. This part of the method seems correct and does not require changes.

Identify the Recursive Call

The current recursive call is \( \text{return} \; n + \text{sum}(n) \). This call is incorrect because calling \( \text{sum}(n) \) means it will keep calling the same \( n \), leading to infinite recursion.

Correct the Recursive Call

To correctly sum the numbers from 0 to \( n \), the recursive call should be made with \( n-1 \). Therefore, the correct recursive call should be \( \text{return} \; n + \text{sum}(n-1) \). This ensures that each recursive call reduces \( n \) until it reaches the base case.

Final Correct Method

The corrected method should be: ```public int sum(int n) { if (n == 0) return 0; else return n + sum(n - 1);}```This method checks if \( n \) is 0 (the base case), and otherwise adds \( n \) to the result of a recursive call with \( n-1 \), ensuring proper recursion.

Key concepts

Base Case

In recursion, the base case is crucial as it defines the stopping point for the recursive calls. For the given problem, the base case is identified as \( n == 0 \). This means, when \( n \) is equal to zero, the function should halt further recursive calls and return the value of zero. The base case acts like an anchor, preventing the recursion from continuing infinitely. It's important because it provides a condition under which the function can terminate. Without it, a recursive function would call itself indefinitely, leading to a program crash. For most recursive functions, ensuring that the base case is correct is half the battle won!

Recursive Call

The recursive call in a recursive function refers to the part of the code where the function calls itself. In the context of this problem, the original line used was \( \text{return} \, n + \text{sum}(n) \). Unfortunately, this line kept calling \( \text{sum}(n) \) without ever reducing the value of \( n \). This results in the function never reaching the base case. Instead, our recursive call should have a decrementing factor, specifically \( \text{sum}(n-1) \). This change means every function call now progresses towards the base case by reducing \( n \) by 1 each time. Correctly structuring your recursive call not only ensures that the function will eventually meet the base case, but also properly computes the desired result by performing the action in a divide-and-conquer fashion.

Infinite Recursion

Infinite recursion is a situation where a recursive function continuously calls itself without ever reaching a base case. This happens when there is no decrement in the recursive call, as was initially written in this exercise. By calling \( \text{sum}(n) \) repeatedly with the same \( n \), the function doesn't narrow down toward the base case and instead remains in an endless loop. To prevent infinite recursion, you must ensure that each recursive call moves closer to the base case. Also, always review your base case conditions. Correctly setting them and ensuring your recursive calls work toward satisfying those conditions is key to avoiding infinite loops.

Summation Method

The summation method in this context refers to recursively adding numbers from a given number \( n \) down to 0. The corrected method uses both a well-defined base case and proper recursive call structure to achieve this. The process starts with a given number \( n \), adds it to the sum of numbers below it through the call \( \text{sum}(n-1) \). When \( n \) finally reaches 0, the base case activates, and \( 0 \) is returned as the stopping result. This means the sum of numbers from 0 to \( n \) builds up gradually through recursive calls. The key takeaway in mastering the summation method lies in understanding each component: the base case for halting and the recursive call for progression, allowing us to effectively compute combinations and accumulations.