Question

Write a recursive function definition for the following function: int squares(int n); //Precondition: n >= 1 //Returns the sum of the squares of numbers 1 through n. For example, squares (3) returns 14 because \(1^{2}+2^{2}+3^{2}\) is 14

Short answer

Question: Write a recursive function for calculating the sum of squares of numbers from 1 to n, where n >= 1. Answer: To write a recursive function for calculating the sum of squares from 1 to n, we first understand the concept of recursion, where a function calls itself to solve a smaller version of the problem. We determine the base case as n = 1, where the sum of squares is 1. We then construct the recursive case, squares(n) = squares(n-1) + n^2, and combine the base case and the recursive case into the following function: ```cpp int squares(int n) { // Base case if (n == 1) { return 1; } // Recursive case return squares(n - 1) + n*n; } ```

Step-by-step solution

Understand recursion

Recursion is a programming concept where a function calls itself to solve a smaller version of the same problem. The function keeps calling itself with simpler inputs until it reaches a base case, which is a simple input for which the output is directly calculated without any further function calls.

Determine the base case

The base case is required for recursion to stop and start returning the accumulated results. Since the precondition is given that n >= 1, we can choose the base case when n = 1. At this point, there's only one square to add up which is 1, so the function should return 1.

Construct the recursive case

For the recursive case, we need to relate the result of the function with input n, to the result of the function with the input (n-1). We know that squares(n) returns the sum of squares till n. So, the sum of squares till n can be expressed as the sum of squares till (n-1) and the square of n: squares(n) = squares(n-1) + n^2.

Combine the base case and recursive case to form the complete solution

Finally, we can now combine the base case and the recursive case to write the complete recursive function: int squares(int n) { // Base case if (n == 1) { return 1; } // Recursive case return squares(n - 1) + n*n; }

Key concepts

recursive function

A recursive function is a function that solves a problem by calling itself. This technique helps break down complex problems into simpler versions, making the problem easier to solve. Think of it like a Russian nesting doll.
Each function call adds a new layer, like a smaller doll within the bigger one, tackling a smaller part of the problem.

• It involves defining the base case: the simplest instance which can be solved without further recursion.
• The function must then call itself to work on the remaining problem, this is the recursive case.

This allows the function to "nest" smaller instances of the problem until reaching one that can be directly solved, at which point the solution is built back up.

base case

In recursion, the base case acts as the stopping condition. It's crucial because it prevents infinite loops by providing a simple condition where the function can return a result without further calling itself.
For our function, when calculating the sum of squares, an appropriate base case is when the input is the smallest possible value, that is, 1.
- If we have the function call `squares(1)`, our base case returns 1, since only one number is squared. - This base case allows the function to know when to stop recursing and begin building the solution back. Having a well-defined base case ensures that our recursive function is both efficient and correctly terminates.

recursive case

The recursive case in a recursive function is where the actual work gets done by breaking down the problem gradually using recursion.
It links the current input to a smaller version of the problem.
In the case of calculating the sum of squares:

• Our function is asked to find the sum of squares from 1 to `n`.
• For `squares(n)`, the recursive case is defined as `squares(n-1) + n^2`.

This pattern continues until it hits the base case, with each call reducing the value of `n` by 1, thus steadily decreasing the problem size.

sum of squares

The sum of squares is a mathematical concept involving adding up the squares of a series of numbers. In this particular recursive exercise, we focus on summing squares from 1 up to `n`.
Here's why the formula works:- Squaring each number in the range, starting from 1 up to `n`, like this: \[1^2 + 2^2 + 3^2 +...+ n^2\]- This represents the sum of the squares of all these numbers, giving a cumulative result.Recursive functions allow us to compute this sum by continually adding a new squared value until reaching 1, which simplifies direct computation and helps efficiently code larger calculations.