Objective: Practice writing recursive functions in python3  Make the five recursive functions described below in python3 by using the starter code recursive_functions.py. For each function, figure out how to solve it conceptually : write down the base case (when recursion stops) and how each recursive function-call moves towards the base case. The functions should not print anything (except you may add temporary print statements to help debug them). You can test your program by using the provided program test_recursive_functions.py. Don't edit the test program. Put it into the same directory (folder) with your recursive_functions.py and run it. It will import your functions as a module, test your functions, and tell you when each function is returning correct results. 1. Factorial  In math, if you have a number n, the factorial function (written n!) computes n x (n-1) x (n-2) x (n-3) x ... x 1. For example: 0! is defined to be 1 1! = 1 2! = 2 x 1=2 3! = 3 x 2 x 1=6 4! = 4 x 3 x 2 x 1 = 24 5! = 5 x 4 x 3 x 2 x 1 = 120 Add your code to the provided function signature so it computes the factorial of the integer it is given. You may not use math.factorial() in your function. 2. Recursively Sum  Make a recursive function that takes an integer argument n and returns the sum of the integers from 1 to n. E.g., if n=10, the function should return 55 (1 + 2 + 3 + . . . 10). 3. Recursively Sum List  Make a recursive function sumlist_recursively(l) that accepts a list of numbers as an argument, and returns the sum of all the numbers in the list. E.g., if the function is passed the list [1, 4, 8, 3, 0, 16], it should return 32. 4. Recursively Multiply  Make a recursive function that multiplies its two integer arguments (x and y) recursively. The function should return the value of x times y. Hint: multiplication can be performed as repeated addition, for example.: 7 x 4 = 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28 5. Recursively Reverse a List  Mke a recursive function that reverses a list. For example, given [1, 2, 3, 4], the function would return this list [4, 3, 2, 1]. It does not print anything.   starter code recursive_function.py: def factorial(n):     pass    # replace this line with your lines of recursive code def sum_recursively(n):     pass    # replace this line with your lines of recursive code def sumlist_recursively(l):     pass    # replace this line with your lines of recursive code def multiply_recursively(n, m):     pass    # replace this line with your lines of recursive code def reverse_recursively(l):     pass    # replace this line with your lines of recursive code test_recursive_functions.py: import recursive_functions import math def main():     # Test factorial     print('Testing factorial.')     assert recursive_functions.factorial(0)  == 1     assert recursive_functions.factorial(1)  == math.factorial(1) == 1     assert recursive_functions.factorial(2)  == math.factorial(2) == 2     assert recursive_functions.factorial(5) == math.factorial(5) == 120     assert recursive_functions.factorial(7) == math.factorial(7) == 5040     print('All tests pass for `factorial` (✿◠‿◠)\n')     # Test sum_recursively     print('Testing sum_recursively.')     assert recursive_functions.sum_recursively(0)  == 0     assert recursive_functions.sum_recursively(1)  == sum(range(1+1)) == 1     assert recursive_functions.sum_recursively(2)  == sum(range(2+1)) == 3     assert recursive_functions.sum_recursively(10) == sum(range(10+1)) == 55     print('All tests pass for `sum_recursively` (✿◠‿◠) ')          # Test sumlist_recursively(l)     print('Testing sumlist_recursively.')     assert recursive_functions.sumlist_recursively([1,2,3]) == sum([1,2,3])     assert recursive_functions.sumlist_recursively([42,16,99,102,1]) == sum([42,16,99,102,1])     assert recursive_functions.sumlist_recursively([17,13,9,5,1]) == sum([17,13,9,5,1])     print('All tests pass for r_sumlist (✿◠‿◠)\n')     # Test multiply_recursively     print('Testing multiply_recursively.')     assert recursive_functions.multiply_recursively(5, 1) == 5*1 == 5     assert recursive_functions.multiply_recursively(7, 4) == 7*4 == 28     print('All tests pass for `multiply_recursively` (✿◠‿◠)\n')     # Test reverse_recursively     print('Testing reverse_recursively.')     assert recursive_functions.reverse_recursively([1, 2, 3, 4]) == [4, 3, 2, 1]     life = ['born', 'grow up', 'grow old']     assert recursive_functions.reverse_recursively(life) == ['grow old', 'grow up', 'born']     print('All tests pass for `reverse_recursively` (✿◠‿◠)\n') main()

Objective: Practice writing recursive functions in python3

 Make the five recursive functions described below in python3 by using the starter code recursive_functions.py. For each function, figure out how to solve it conceptually : write down the base case (when recursion stops) and how each recursive function-call moves towards the base case. The functions should not print anything (except you may add temporary print statements to help debug them).

You can test your program by using the provided program test_recursive_functions.py. Don't edit the test program. Put it into the same directory (folder) with your recursive_functions.py and run it. It will import your functions as a module, test your functions, and tell you when each function is returning correct results.

1. Factorial 
In math, if you have a number n, the factorial function (written n!) computes

n x (n-1) x (n-2) x (n-3) x ... x 1. For example:

• 0! is defined to be 1

• 1! = 1

• 2! = 2 x 1=2

• 3! = 3 x 2 x 1=6

• 4! = 4 x 3 x 2 x 1 = 24

• 5! = 5 x 4 x 3 x 2 x 1 = 120

  Add your code to the provided function signature so it computes the factorial of the integer it is given. You may not use math.factorial() in your function.

2. Recursively Sum 
Make a recursive function that takes an integer argument n and returns the sum of the

integers from 1 to n. E.g., if n=10, the function should return 55 (1 + 2 + 3 + . . . 10).

3. Recursively Sum List 

Make a recursive function sumlist_recursively(l) that accepts a list of numbers as an argument, and returns the sum of all the numbers in the list. E.g., if the function is passed the list [1, 4, 8, 3, 0, 16], it should return 32.

4. Recursively Multiply 

Make a recursive function that multiplies its two integer arguments (x and y) recursively. The function should return the value of x times y. Hint: multiplication can be performed as repeated addition, for example.:

7 x 4 = 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28

5. Recursively Reverse a List 
Mke a recursive function that reverses a list. For example, given [1, 2, 3, 4], the function

would return this list [4, 3, 2, 1]. It does not print anything.

starter code recursive_function.py:

def factorial(n):
    pass    # replace this line with your lines of recursive code

def sum_recursively(n):
    pass    # replace this line with your lines of recursive code

def sumlist_recursively(l):
    pass    # replace this line with your lines of recursive code

def multiply_recursively(n, m):
    pass    # replace this line with your lines of recursive code

def reverse_recursively(l):
    pass    # replace this line with your lines of recursive code

test_recursive_functions.py:

import recursive_functions
import math

def main():
    # Test factorial
    print('Testing factorial.')
    assert recursive_functions.factorial(0)  == 1
    assert recursive_functions.factorial(1)  == math.factorial(1) == 1
    assert recursive_functions.factorial(2)  == math.factorial(2) == 2
    assert recursive_functions.factorial(5) == math.factorial(5) == 120
    assert recursive_functions.factorial(7) == math.factorial(7) == 5040
    print('All tests pass for `factorial` (✿◠‿◠)\n')

    # Test sum_recursively
    print('Testing sum_recursively.')
    assert recursive_functions.sum_recursively(0)  == 0
    assert recursive_functions.sum_recursively(1)  == sum(range(1+1)) == 1
    assert recursive_functions.sum_recursively(2)  == sum(range(2+1)) == 3
    assert recursive_functions.sum_recursively(10) == sum(range(10+1)) == 55
    print('All tests pass for `sum_recursively` (✿◠‿◠) ')
    
    # Test sumlist_recursively(l)
    print('Testing sumlist_recursively.')
    assert recursive_functions.sumlist_recursively([1,2,3]) == sum([1,2,3])
    assert recursive_functions.sumlist_recursively([42,16,99,102,1]) == sum([42,16,99,102,1])
    assert recursive_functions.sumlist_recursively([17,13,9,5,1]) == sum([17,13,9,5,1])
    print('All tests pass for r_sumlist (✿◠‿◠)\n')

    # Test multiply_recursively
    print('Testing multiply_recursively.')
    assert recursive_functions.multiply_recursively(5, 1) == 5*1 == 5
    assert recursive_functions.multiply_recursively(7, 4) == 7*4 == 28
    print('All tests pass for `multiply_recursively` (✿◠‿◠)\n')

    # Test reverse_recursively
    print('Testing reverse_recursively.')
    assert recursive_functions.reverse_recursively([1, 2, 3, 4]) == [4, 3, 2, 1]
    life = ['born', 'grow up', 'grow old']
    assert recursive_functions.reverse_recursively(life) == ['grow old', 'grow up', 'born']
    print('All tests pass for `reverse_recursively` (✿◠‿◠)\n')

main()