Tutorial#

Similarly to the previous chapter, you will use a computer to gain numerical evidence for a problem.

Problem

The Fibonacci numbers are defined by the following sequence:

\[\begin{split} \left\{\begin{array}{l} a_0 = 0,\\ a_1 = 1,\\ a_n = a_{n - 1} + a_{n - 2}, n \geq 2\end{array}\right. \end{split}\]

Verify that the following identity holds for \(n\leq 500\):

\[ \sum_{i=0}^n a_i = a_{n + 2} - 1 \]

You will start by defining a function for \(a(n)\):

import functools


@functools.lru_cache()
def get_fibonacci(n):
    """
    A function to give the nth Fibonacci number using the recursive
    definition.

    Note that this also uses a cache.

    Parameters
    ----------
    n: int
        The index of the Fibonacci number

    Returns
    -------
    int
        The nth Fibonacci number
    """
    if n == 0:
        return 0
    if n == 1:
        return 1
    return get_fibonacci(n - 1) + get_fibonacci(n - 2)

Attention

This uses caching in that function definition with lru_cache. This is not necessary but makes the code more efficient. Caching is covered in What is caching?.

You will print the first 10 numbers to ensure everything is working correctly:

for n in range(10):
    print(get_fibonacci(n))

Now write a function that returns a boolean: True if the equation holds for a given value of \(n\), False otherwise.

def check_theorem(n):
    """
    A function that generate the lhs and rhs of the
    following relationship:

    \sum_{i=0}^n a_i = a_{n + 2} - 1

    Where `a_i` is the i-th Fibonacci number.

    It checks if the relationship holds.

    Parameters
    ----------
    n: int
        The index n for which the theorem is to be verified.

    Returns
    -------
    bool
        Whether or not the theorem holds for a given n.
    """
    sum_of_fibonacci = sum(get_fibonacci(i) for i in range(n + 1))
    return sum_of_fibonacci == get_fibonacci(n + 2) - 1

Generate checks for \(n\leq 500\):

checks = [check_theorem(n) for n in range(501)]
checks

[True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
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 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
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 True,
 True,
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 True,
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 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
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 True,
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 True,
 True,
 True,
 True,
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 True,
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 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True,
 True]

Confirm that all the booleans in checks are True:

all(checks)

True

Python for mathematics

Tutorial

Tutorial#