Sequences

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 $a_1, a_2, a_3, …$ is defined by:

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

where $k$ is a constant.

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

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 $k$ as a symbol:

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

To find $a_3$:

>>> 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)
{20441/10220}

Come back: with time take any questions.

Point at resources.