Recursion A useful feature of user-defined functions is recursion, the ability of a function to call itself. For example, consider the following definition' of the factorial \(n !\) of a positive integer \(n\) : $$ n != \begin{cases}1 & \text { if } n=1 \\ n \times(n-1) ! & \text { if } n>1\end{cases} $$ This constitutes a complete definition of the factorial which allows us to calculate the value of \(n !\) for any positive integer. We can employ this definition directly to create a Python function for factorials, like this: def factorial \((\mathrm{n}):\) if \(\mathrm{n}==1:\) return 1 else: return \(\mathrm{n} *\) factorial \((\mathrm{n}-1)\) Note how, if \(n\) is not equal to 1 , the function calls itself to calculate the factorial of \(n-1\). This is recursion. If we now say "print (factorial (5))" the computer will correctly print the answer 120 .

Step by step solution

01

Understand the Problem Definition

The problem is to calculate the factorial of a positive integer using recursion. The definition of factorial is given as: 1. If \(n = 1\), then \(n! = 1\) 2. If \(n > 1\), then \(n! = n \times (n - 1)!\)

02

Recursion Explanation

Recursion is a process where a function calls itself. Here, the factorial function calls itself with the argument \(n-1\) until the base case, \(n=1\), is reached.

03

Python Function Definition

The given Python function is defined as follows: ```python def factorial(n): if n == 1: return 1 else: return n * factorial(n-1) ``` This function works by checking if \(n\) is 1. If true, it returns 1. Otherwise, it recursively calls itself with \(n-1\) and multiplies the result by \(n\).

04

Execute the Function

To calculate the factorial of 5, we call the function as follows: ```python print(factorial(5)) ``` This will compute \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

05

Detailed Execution of factorial(5)

Here is the step-by-step breakdown of calculating \(5!\): 1. factorial(5) = 5 * factorial(4) 2. factorial(4) = 4 * factorial(3) 3. factorial(3) = 3 * factorial(2) 4. factorial(2) = 2 * factorial(1) 5. factorial(1) = 1 Combining these steps, we get \(5! = 5 * 4 * 3 * 2 * 1 = 120\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factorial function

The factorial function is a mathematical function that multiplies a number by every integer less than it down to 1. It's represented by an exclamation mark (!) after the number. For example, the factorial of 5 (written as 5!) is equal to 5 × 4 × 3 × 2 × 1. This gives us the value 120.
Factorials are commonly used in permutations, combinations, and other areas of mathematics.
The general rule for factorials is:

• If \(n = 1\), then \(n! = 1\)
• If \(n > 1\), then \(n! = n \times (n - 1)!\)

This second rule shows the recursive nature of the factorial function, as it refers to the function itself for its calculation.

Python functions

Python functions are blocks of reusable code designed to perform specific tasks. They help in organizing code and making it more modular.
A Python function is defined using the 'def' keyword, followed by the function name and parentheses (). Here is the syntax to define a function:
```python
def function_name(parameters):
# code block
return result
```
In our factorial example, the function is defined as follows:
```python
def factorial(n):
if n == 1:
return 1
else:
return n * factorial(n-1)
```
This function checks if the number is 1. If true, it returns 1. Otherwise, it calls itself with \(n-1\) and multiplies the result by \(n\).
Notice the use of 'if' and 'else' to handle different cases.

recursive algorithms

A recursive algorithm is an algorithm that solves a problem by solving smaller instances of the same problem. This process continues until it hits a base case, which is solved directly without further recursion.
Key components of a recursive algorithm are:

• Base case: The condition under which recursion ends.
• Recursive step: The process of reducing the problem into smaller instances and calling the function itself.

In the factorial function example, the base case is when \(n = 1\), returning 1. The recursive step is when the function calls itself with \(n - 1\).
Recursive algorithms are powerful but can lead to stack overflow if not handled correctly, as each function call adds a new layer to the call stack. Always ensure that recursive algorithms have well-defined base cases to avoid infinite recursion.