5 A doctors' surgery starts a campaign to reduce missed appointments.
The number of missed appointments for each of the first five weeks after the start of the campaign is shown below. This data could be modelled by an equation of the form \(y = p q ^ { x }\) where \(p\) and \(q\) are constants.

1. Show that this relationship may be expressed in the form \(\log _ { 10 } y = m x + c\), expressing \(m\) and \(c\) in terms of \(p\) and/or \(q\). The diagram below shows \(\log _ { 10 } y\) plotted against \(x\), for the given data.
   \includegraphics[max width=\textwidth, alt={}, center]{54bdddcb-c1ef-4b60-af6c-ac944cae29fe-05_737_1668_1233_258}
2. Estimate the values of \(p\) and \(q\).
3. Use the model to predict when the number of missed appointments will fall below 20. Explain why this answer may not be reliable.
4. A student suggests that, for any prime number between 20 and 40, when its digits are squared and then added, the sum is an odd number. For example, 23 has digits 2 and 3 which gives \(2 ^ { 2 } + 3 ^ { 2 } = 13\), which is odd. Show by counter example that this suggestion is false.
5. Prove that the sum of the squares of any three consecutive positive integers cannot be divided by 3 .