Question

Consider the following method: public static int JPS(int n) { if(n==0)return 0; return n + JPS(n-1); } What does this method compute?

Short answer

The method computes the sum of the first `n` natural numbers.

Step-by-step solution

Understanding the Method

The method `JPS` takes an integer `n` and checks if it is equal to 0. If `n` is 0, it returns 0. Otherwise, it returns the sum of `n` and the result of `JPS(n-1)`. This pattern resembles the definition of a cumulative sum of natural numbers.

Recognizing the Pattern

The method essentially adds each number from `n` down to 1 recursively. To understand its output, let's consider a few examples. For `JPS(3)`, it computes `3 + JPS(2)`, then `2 + JPS(1)`, and finally `1 + JPS(0)`, which leads to `3 + 2 + 1 = 6`. This is the sum of the first 3 positive integers.

Defining the Computation

The method calculates the sum of all natural numbers from 1 to `n`. This is equivalent to the mathematical expression for the sum of the first `n` natural numbers: \[ S = \frac{n(n+1)}{2} \] where `S` is the sum. Each recursive call handles one number until it reaches 0.

Key concepts

Cumulative Sum

A cumulative sum refers to the total of a sequence of numbers where each number is added one after the other. In the context of our exercise, the `JPS` method demonstrates this concept by adding each natural number from `n` down to 1. This results in what we call a cumulative sum.

• The sequence of additions starts with the highest number (`n`) and proceeds down to 1.
• This method uses recursion, where the function calls itself to continue the sum until it reaches the base case (`n = 0`).
• The outcome is the summation of all these numbers, which forms the cumulative sum up to `n`.

Understanding cumulative sums through recursive functions like these helps students realize how mathematical concepts can be expressed programmatically.

Natural Numbers

Natural numbers are a set of positive integers starting from 1, 2, 3, and so on. They do not include zero in classical definitions, but they do include all positive integers. In our recursive function `JPS`, we work exclusively with natural numbers.

• The process begins at a given natural number `n` and counts down to 1.
• The operation excludes negative numbers, as recursion stops once `n` becomes 0.
• The concept of natural numbers is vital because the sum only makes sense when we add positive whole numbers.

This method highlights the properties of natural numbers, which are the basic building blocks of arithmetic calculations.

Mathematical Expressions

Mathematical expressions allow us to represent numbers and operations in a concise manner. The recursive function `JPS` can be described through a well-known mathematical formula for the sum of the first `n` natural numbers: \[S = \frac{n(n+1)}{2}\]This expression simplifies the entire summation process by giving a direct result without recursion.

• For instance, to find the sum of the first 3 natural numbers, we substitute `n = 3` into the expression and get 6.
• Such expressions are vital in simplifying computations and gaining insights into the nature of these sums.
• Using this expression alongside a recursive function broadens the understanding of how numbers can be manipulated mathematically.

Knowing how to convert recursive logic into a straightforward expression is a skill that illustrates the power of mathematical thinking.