Tutorial#

We will solve the following problem using a computer using a programming technique called recursion.

Problem

A sequence \(a_1, a_2, a_3, …\) is defined by:

\[\begin{split} \left\{ \begin{array}{l} a_1 = k,\\ a_{n + 1} = 2a_n – 7, n \geq 1, \end{array} \right. \end{split}\]

where \(k\) is a constant.

  1. Write down an expression for \(a_2\) in terms of \(k\).

  2. Show that \(a_3 = 4k -21\)

  3. Given that \(\sum_{r=1}^4 a_r = 43\) find the value of \(k\).

We will use a Python to define a function that reproduces the mathematical definition of \(a_k\):

def generate_a(k_value, n):
    """
    Uses recursion to return a_n for a given value of k:

    a_1 = k
    a_n = 2a_{n-1} - 7
    """
    if n == 1:
        return k_value
    return 2 * generate_a(k_value, n - 1) - 7

Attention

This is similar to the mathematical definition the Python definition of the function refers to itself.

We can use this to compute \(a_3\) for \(k=4\):

generate_a(k_value=4, n=3)

-5

We can use this to compute \(a_5\) for \(k=1\):

generate_a(k_value=1, n=5)

-89

Finally it is also possible to pass a symbolic value to k_value. This allows us to answer the first question:

import sympy as sym

k = sym.Symbol("k")
generate_a(k_value=k, n=2)
\[\displaystyle 2 k - 7\]

Likewise for \(a_3\):

generate_a(k_value=k, n=3)
\[\displaystyle 4 k - 21\]

For the last question we start by computing the sum:

\[ \sum_{r=1}^4 a_r = 43 \]
sum_of_first_four_terms = sum(generate_a(k_value=k, n=r) for r in range(1, 5))
sum_of_first_four_terms
\[\displaystyle 15 k - 77\]

This allows us to create the given equation and solve it:

equation = sym.Eq(sum_of_first_four_terms, 43)
sym.solveset(equation, k)
\[\displaystyle \left\{8\right\}\]

Important

In this tutorial we have

  • Defined a function using recursion.

  • Called this function using both numeric and symbolic values.