Question

The Fibonacci sequence starts \(1,1,2,3,5,8, \ldots .\) Each number in the sequence (after the first two) is the sum of the previous two. Write a program that computes and outputs the \(n\) th Fibonacci number, where \(n\) is a value entered by the user.

Short answer

Iteratively compute Fibonacci numbers using a loop and output the result for the user's input value.

Step-by-step solution

Understand the Fibonacci sequence

The Fibonacci sequence is a series of numbers in which each number after the first two is the sum of the two preceding ones. The sequence starts as follows: 1, 1, 2, 3, 5, 8, and so on. Each term can be found using the formula: \[ F(n) = F(n-1) + F(n-2) \] where \( F(n) \) is the \( n \)-th Fibonacci number and \( F(1) = F(2) = 1 \).

Define a function to compute Fibonacci numbers

Create a function, `fibonacci(n)`, that will compute the Fibonacci sequence's \( n \)-th number. This function will use an iterative approach to avoid the inefficiency of recursive calls for large \( n \).

Implement the iterative algorithm

Start with two initial numbers, \( a = 1 \) and \( b = 1 \), corresponding to \( F(1) \) and \( F(2) \). For \( n > 2 \), use a loop to calculate the next Fibonacci number as follows:1. Loop from 3 to \( n \).2. In each iteration, calculate the sum \( c = a + b \).3. Update the values of \( a \) and \( b \) so that \( a = b \) and \( b = c \).4. After the loop ends, \( b \) will contain the \( n \)-th Fibonacci number.

Get user input and output the result

Allow the user to input a positive integer \( n \). Use the `input()` function in Python to retrieve this value. Pass this value to the `fibonacci` function and print out the resulting \( n \)-th Fibonacci number.

Write the complete program

Below is a complete Python program to achieve the solution:```pythondef fibonacci(n): if n <= 0: return 'Error: Input must be a positive integer.' elif n == 1 or n == 2: return 1 else: a, b = 1, 1 for _ in range(3, n + 1): a, b = b, a + b return bn = int(input('Enter a positive integer to find the n-th Fibonacci number: '))print(f'The {n}-th Fibonacci number is {fibonacci(n)}')```This program correctly calculates the \( n \)-th Fibonacci number given a user's input.

Key concepts

Iterative Algorithm

The Fibonacci sequence is a classic example of where iterative algorithms shine. Unlike recursive methods, which call themselves repeatedly, iterative algorithms use loops to achieve their results. This approach is particularly advantageous when calculating Fibonacci numbers because it significantly decreases the amount of memory and processing time required for larger values of \( n \). An iterative algorithm can handle this efficiently by:

• Initializing with base values: Start with the first two numbers of the sequence, \( a = 1 \) and \( b = 1 \), corresponding to \( F(1) \) and \( F(2) \).
• Using a loop: Continue from \( n = 3 \) to the desired \( n \)-th number. Each loop iteration calculates the next Fibonacci number.
• Updating variables: Instead of recalling values, simply update \( a \) and \( b \) with each iteration, where \( a = b \), and \( b = c \).
• Returning the result: Once the loop completes, the variable \( b \) will contain the \( n \)-th Fibonacci number.

This use of variables and a simple loop ensures that each number in the sequence is computed quickly and without unnecessary complexity.

Function Definition

Defining functions in programming is about setting up reusable blocks of code that perform a specific task. In this instance, we created a function called `fibonacci(n)` that calculates the \( n \)-th Fibonacci number. Defining this function involves specifying the logic the function will employ to do its job.Here's what happens within the `fibonacci(n)` function:

• Parameter setup: The function takes one argument, \( n \), which represents the position in the Fibonacci sequence we're interested in.
• Input validation: To ensure the function operates correctly, a check is performed to validate that \( n \) is a positive integer. If not, it returns an error message.
• Base cases: If \( n \) is 1 or 2, the function returns 1 immediately, as these are the initial values in the sequence.
• Iterative calculation: For all other values, an iterative approach is employed to calculate the Fibonacci number efficiently.
• Final output: Once calculated, the function returns the \( n \)-th Fibonacci number.

Function definitions like this provide clarity, allowing the main program to call this process without worrying about the underlying complexity.

User Input Handling

Handling user input is a crucial aspect of interactive programs. It's how the program communicates with its users to receive necessary information. In calculating the Fibonacci sequence, we need to determine which Fibonacci number to find. This is where the user input comes into play. Here's how user input handling works in this program:

• Prompt the user: Use the `input()` function to ask the user to enter a positive integer.
• Convert the input: The user's input is typically a string, so it's converted to an integer using the `int()` function.
• Validate the input: Although basic validation is done in the `fibonacci()` function, it's good practice to screen inputs before processing. Ensure that the user provides a positive integer.
• Pass the value to the function: The validated input is then used to retrieve the desired Fibonacci number by calling the `fibonacci()` function.
• Output the result: Finally, the program uses the `print()` function to display the result back to the user.

By correctly implementing these steps, we ensure that the program is user-friendly and can handle various inputs without crashing.