Question

A Fibonacci sequence is a sequence of numbers where each successive number is the sum of the previous two. The classic Fibonacci sequence begins: \(1,1,2,3,5,8,13, \ldots .\) Write a program that computes the \(n\) th Fibonacci number where \(n\) is a value input by the user. For example, if \(n=6,\) then the result is 8.

Short answer

Use a loop to calculate the nth Fibonacci number, updating two variables iteratively.

Step-by-step solution

Understand the Fibonacci Sequence

The Fibonacci sequence is defined such that each number is the sum of the two preceding ones, typically starting with the numbers 1 and 1. So, it begins as follows: 1, 1, 2, 3, 5, 8, 13, ... .

Initialize Variables

To find the nth Fibonacci number, we need to keep track of two variables: the last two numbers in the sequence. Start by setting `a = 1` and `b = 1`, since the first two Fibonacci numbers are both 1.

Iterate to Calculate the Sequence

Use a loop to calculate the Fibonacci numbers until you reach the nth number. In each iteration, calculate the next Fibonacci number `c = a + b`, then update `a = b` and `b = c`. Repeat this process from the third element to the nth element.

Implement the Loop in Code

Write a Python loop that starts from the third Fibonacci number up to the nth number: ```python n = int(input("Enter n: ")) a, b = 1, 1 for i in range(2, n): c = a + b a = b b = c print(b) ``` This code snippet will compute the nth Fibonacci number.

Verify the Output

Once the loop finishes, `b` will hold the nth Fibonacci number. For example, if n=6, the output should be 8, confirming that the program correctly calculates the sequence.

Key concepts

Python Programming

Python is a powerful and easy-to-learn programming language that is perfect for implementing algorithms like the Fibonacci sequence. Its syntax is clear and readable, which makes it accessible for beginners and experienced developers alike. In Python, you can quickly write code to accomplish complex tasks using simple instructions. The Fibonacci algorithm can be implemented with just a few lines of Python code, demonstrating Python's efficiency and elegance.
Using Python, we can handle large calculations and manage iterations smoothly, which are essential for computing sequences like Fibonacci. Python provides several built-in functions and data types that make these operations straightforward.
One of the strong points of Python is its dynamic typing system, which means you don't need to declare the type of a variable; Python handles it automatically. In the Fibonacci program, you can easily swap and update variable values without worrying about type compatibility.

Iteration

Iteration is a fundamental concept in programming that involves repeating a set of instructions until a particular condition is met. This concept is essential for generating the Fibonacci sequence up to the nth term. By using iteration, we can efficiently compute Fibonacci numbers by cycling through calculation steps repeatedly.
In Python, iteration is typically achieved using loops such as `for` and `while`. In the Fibonacci program, a `for` loop is used to iterate through the steps needed to compute each number in the sequence starting from the third number. The `range` function helps determine how many times the loop should run.
By iterating with a loop, you can efficiently calculate each subsequent number by adding the last two numbers in the sequence. This approach avoids unnecessary recalculations, making it a much more efficient way to compute Fibonacci numbers compared to a straightforward recursive approach.

Variables

Variables are named storage locations in programming that hold data. They're essential in any programming task, including computing Fibonacci numbers, as they allow you to store and manipulate data temporarily during execution. In Python, variables are dynamically typed, meaning they can change types during runtime.
In the Fibonacci sequence program, two crucial variables, `a` and `b`, are used to keep track of the last two values in the sequence. These variables are initially set to 1, corresponding to the first two Fibonacci numbers. As the program iterates, these variables are updated to hold the next numbers in the sequence.
Using variables effectively ensures that the program correctly adds the previous numbers to compute the next Fibonacci number. Variables make our code more flexible and reusable because we can easily adjust what values they hold.

User Input

User input is a fundamental component in many programming tasks, allowing programs to gather information from the user. In Python, user input can be captured using the `input()` function. This function reads a line from input, typically entered by a user, and returns it as a string. The original Fibonacci program utilizes user input to determine how many numbers in the sequence should be computed (`n`). The input received is converted into an integer using `int()`, which is necessary because operations like `range()` require an integer type. Allowing the user to input a value makes the Fibonacci program dynamic and interactive. Instead of calculating a fixed sequence length, the program can adapt to compute up to any Fibonacci number specified by the user. This interaction is fundamental in creating useful and flexible software solutions.