8. The owner of a new company models the number of customers that the company will have at the end of each month. The owner assumes that

• a constant proportion, \(p\) (where \(0 < p < 1\) ), of the previous month's customers will be retained for the next month
• a constant number of new customers, \(k\), will be added each month.

Let \(u _ { n }\) (where \(n \geqslant 1\) ) represent the number of customers that the company will have at the end of \(n\) months. The company has 5000 customers at the end of the first month.

1. By setting up a first order recurrence relation for \(u _ { n + 1 }\) in terms of \(u _ { n }\), determine an expression for \(u _ { n }\) in terms of \(n , p\) and \(k\). The owner believes that \(95 \%\) of the previous month’s customers will be retained each month and that there will be 10000 new customers each month. According to the model, the company will first have at least 135000 customers by the end of the \(m\) th month.
2. Using logarithms, determine the value of \(m\). Please check the examination details below before entering your candidate information \section*{Further Mathematics} Advanced
   PAPER 4D: Decision Mathematics 2 \section*{Answer Book} Do not return the question paper with the answer book.
   1.

   	1	2	3	4
   A	32	45	34	48
   B	37	39	50	46
   C	46	44	40	42
   D	43	45	48	52

   You may not need to use all of these tables.

   	1	2	3	4
   A
   B
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   D

   	1	2	3	4
   A
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   	1	2	3	4
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   VALV SIHI NI IIIIIM ION OC
   VIAV SIUL NI JAIIM ION OC
   VIAV SIHI NI III IM ION OC

   	1	2	3	4
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   	1	2	3	4
   \(A\)
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   	1	2	3	4
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   	1	2	3	4
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   2.

   VAMV SIHI NI IIIHM ION OO

   VIAV SIHI NI JIIIM I ON OC

   VJYV SIHI NI JIIIM ION OC

   3. 4. \begin{figure}[h]
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   \end{figure} 5.

3. 	\(R\)	\(S\)	\(T\)
   \(A\)
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   	\(R\)	\(S\)	\(T\)
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   	\(R\)	\(S\)	\(T\)
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   6.

   Stage	State	Action	Destination	Value
   May	2	0	0	160
   	1	1	0	80 + 35
   	0	2	0	70

   Stage	State	Action	Destination	Value

   Month	January	February	March	April	May
   Number made

   Minimum cost: \(\_\_\_\_\)
   7. Player A \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Player B}

   	Option W	Option X	Option Y	Option Z
   Option Q	4	3	-1	-2
   Option R	-3	5	-4	\(k\)
   Option S	-1	6	3	-3

   \end{table} The tableau for (d)(ii) can be found at the bottom of page 16

   b.v.				□	□	□	□		Value
   \includegraphics[max width=\textwidth, alt={}]{cea07472-f93b-4a7b-b362-89fb8c0af4a9-24_56_77_2348_182}		□	□
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   8.