Question

Write definitions for the following two functions: \(\operatorname{sum} N(n)\) returns the sum of the first n natural numbers. sumNCubes(n) returns the sum of the cubes of the first n natural numbers. Then use these functions in a program that prompts a user for an \(n\) and prints out the sum of the first \(n\) natural numbers and the sum of the cubes of the first \(n\) natural numbers.

Short answer

Define functions for sum and cubes, then use them to print results.

Step-by-step solution

Define the Function sumN

The function \( \text{sumN}(n) \) calculates the sum of the first \( n \) natural numbers. The formula for this sum is \( \frac{n(n + 1)}{2} \). This function takes an integer \( n \) as input and returns the calculated sum using the aforementioned formula.

Define the Function sumNCubes

The function \( \text{sumNCubes}(n) \) calculates the sum of the cubes of the first \( n \) natural numbers. The formula for this sum is \( \left( \frac{n(n + 1)}{2} \right)^2 \). This function takes an integer \( n \) as input and returns the calculated sum of cubes using this formula.

Implement the Program

Write a program that prompts the user to enter a number \( n \). The program should then use the defined functions \( \text{sumN}(n) \) and \( \text{sumNCubes}(n) \) to calculate both the sum and the sum of cubes of the first \( n \) natural numbers. Finally, print out these calculated values to the user.

Code Example

Here is a simple Python code that demonstrates the above logic: ```python def sumN(n): return n * (n + 1) // 2 def sumNCubes(n): return (n * (n + 1) // 2) ** 2 n = int(input('Enter a number: ')) print(f'Sum of the first {n} natural numbers: {sumN(n)}') print(f'Sum of the cubes of the first {n} natural numbers: {sumNCubes(n)}') ```

Key concepts

sum of natural numbers

The sum of natural numbers is a basic yet important concept in mathematics. It involves calculating the sum of all natural numbers up to a particular number \( n \). Natural numbers are the sequence of positive integers starting from 1, like 1, 2, 3, and so on.
To find the sum of the first \( n \) natural numbers, you can use the formula:

\[ \text{Sum} = \frac{n(n + 1)}{2} \]

This formula is derived from the principle of arithmetic series, which simplifies the process by providing a quick computational shortcut.
For example, if you want to find the sum of the first 5 natural numbers (1 + 2 + 3 + 4 + 5), you could individually add each number, but using the formula \( \frac{5(5 + 1)}{2} \) gives you the result 15 directly.

sum of cubes of natural numbers

The sum of cubes of natural numbers is a slightly more complex calculation than just the sum of natural numbers. Instead of adding the numbers themselves, this involves adding the cubes of the numbers.
The formula to calculate the sum of the cubes of the first \( n \) natural numbers is:

\[ \text{Sum of cubes} = \left( \frac{n(n + 1)}{2} \right)^2 \]

This means that you first calculate the sum of numbers up to \( n \), using the formula for the sum of natural numbers, and then square the result.
For instance, to calculate the sum of the cubes of the first 3 natural numbers (1 cube + 2 cube + 3 cube), you would first add the numbers using the formula \( \frac{3(3 + 1)}{2} = 6 \), and then square 6 to get 36.
This relationship elegantly showcases how closely linked arithmetic sequences and their cubes are through mathematics.

functions in Python

In Python programming, functions are reusable blocks of code that perform a specific task. Functions help break down complex problems into simpler pieces, which can be reused throughout different parts of a program.
Defining a function in Python is simple and is done using the 'def' keyword. After defining, you specify the parameters that the function will take as inputs, perform some operations, and return an output.
For example, the function to calculate the sum of natural numbers looks like this:

```pythondef sumN(n): return n * (n + 1) // 2```

This function takes an integer \( n \) and returns the result after applying the formula for the sum of natural numbers. The double slash (//) ensures the operation performs integer division, which is perfect for summing integers.
Similarly, the function to compute the sum of cubes:

```pythondef sumNCubes(n): return (n * (n + 1) // 2) ** 2```

Here, after calculating the sum of \( n \), it squares the result to find the sum of cubes.
Using functions in Python makes your code more readable, organizes it neatly, and allows for easy testing and debugging.