Recurrence relation asymptotic behaviour

3 In the first week of an outbreak of influenza, 9 patients were diagnosed with the virus at a medical practice in Pencaster. Records were kept of \(y\), the total number of patients diagnosed with influenza in week \(n\). The data are shown in Fig. 3. \begin{table}[h] \end{table}

1. Complete the difference table in the Printed Answer Booklet.
2. Explain why a cubic model is appropriate for the data.
3. Use Newton's method to find the interpolating polynomial of degree 3 for these data. In both week 6 and week 7 there were 145 patients in total diagnosed with influenza at the medical practice.
4. Determine whether the model is a good fit for these data.
5. Determine the maximum number of weeks for which the model could possibly be valid.

4. A sequence \(\left\{ u _ { n } \right\}\), where \(n \geqslant 1\), satisfies the recurrence relation $$2 u _ { n } = u _ { n - 1 } - k n ^ { 2 } \text { where } 4 u _ { 2 } - u _ { 0 } = 27 k ^ { 2 }$$ and \(k\) is a non-zero constant.
Show that, as \(n\) becomes large, \(u _ { n }\) can be approximated by a quadratic function of the form \(a n ^ { 2 } + b n + c\) where \(a , b\) and \(c\) are constants to be determined. Please check the examination details below before entering your candidate information
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Candidate Number Level 3 GCE \includegraphics[max width=\textwidth, alt={}, center]{a9f21789-1c5b-42f5-9c5a-3b29d9346c46-05_122_433_356_991}
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□ \section*{Thursday 14 May 2020} You may not need to use all of these tables. 3. \begin{table}[h] \end{table} 4. .

4. A sequence \(\left\{ u _ { n } \right\}\), where \(n \geqslant 0\), satisfies the recurrence relation $$u _ { n + 1 } = \frac { 3 } { 2 } u _ { n } - 2 n ^ { 2 } - 4 \quad u _ { 0 } = k$$ where \(k\) is an integer.

1. Determine an expression for \(u _ { n }\) in terms of \(n\) and \(k\).
   (6) Given that \(u _ { 10 } > 5000\)
2. determine the minimum possible value of \(k\).
   (2)