Sequences week
==============

First meeting
-------------

After this meeting students should:

- Understand how to use define function recursively
- Know what they need to do to prepare for their seventh tutorial.

Problem
*******

Explain to students that we will be solving the following problem:

A sequence :math:`a_1, a_2, a_3, …` is defined by:

.. math::

    \left\{
    \begin{array}{l}
        a_1 = k,\\
        a_{n + 1} = 3a_n – 4, n \geq 1,
    \end{array}
    \right.

where :math:`k` is a constant.


1. Write down an expression for :math:`a_2` in terms of :math:`k`.
2. Show that :math:`a_3 = 9k - 16`
3. Given that :math:`\sum_{r=1}^{12} a_r = 50` find the value of :math:`k`.


Solution
********

Group exercise (breakout rooms of 3): ask students to spend 5 minutes writing a
plan to tackle that problem (not necessarily carrying out each step).

Now explain that we essentially need to add to our tools the ability to define a
function recursively which we can do::

    >>> def generate_a(k_value, n):
    ...     """
    ...     Uses recursion to return a_n for a given value of k:
    ...     a_1 = k
    ...     a_n = 3a_n - 4
    ...     """
    ...     if n == 1:
    ...         return k_value
    ...     return 3 * generate_a(k_value, n - 1) - 4

Spend some time explaining what is happening here. Specifically highlighting
that the programming recursion matches the mathematical definition.

Show how we can make some calls of the function::

    >>> k = 5
    >>> generate_a(k_value=k, n=1)
    5
    >>> generate_a(k_value=k, n=2)
    11
    >>> generate_a(k_value=k, n=3)
    29
    >>> generate_a(k_value=k, n=4)
    83
    >>> generate_a(k_value=k, n=5)
    245

To be able to answer the questions we need :math:`k` as a symbol::

    >>> import sympy as sym
    >>> k = sym.Symbol("k")
    >>> generate_a(k_value=k, n=2)
    3*k - 4

To find :math:`a_3`::

    >>> generate_a(k_value=k, n=3)
    9*k - 16

We now compute the sum of the first 12 terms::


    >>> sum_of_first_12_terms = sum(generate_a(k_value=k, n=r) for r in range(1, 13))
    >>> sum_of_first_12_terms
    265720*k - 531416

Our equation is thus::

    >>> equation = sym.Eq(sum_of_first_12_terms, 50)
    >>> equation
    Eq(265720*k - 531416, 50)
    >>> sym.solveset(equation, k)
    FiniteSet(20441/10220)

Come back: with time take any questions.

Point at resources.

After class email
-----------------

Send the following email after class::

    Hi all,

    A recording of today's class is available at <>.

    In this class I went over a demonstration of using recursion in Python to be
    able to define sequences recursively.

    In preparation for your tutorial tomorrow please work through the seventh
    chapter of the Python for mathematics book:
    https://vknight.org/pfm/tools-for-mathematics/07-sequences/introduction/main.html

    Please get in touch if I can assist with anything,
    Vince