Recurrence relation solution

4. A village has an expected population growth rate (birth rate minus death rate) of \(r \%\) per year. In addition, \(N\) people are expected to move into the village each year. The expected population of the village is modelled by $$u _ { n + 1 } = 1.02 u _ { n } + 50$$ where \(u _ { n }\) is the expected population of the village \(n\) years from now.

1. State
   1. the value of \(r\),
   2. the value of \(N\). Given that the population 1 year from now is expected to be 560
2. solve the recurrence relation for \(u _ { n }\)
3. Hence determine, using algebra, the number of years from now when the model predicts that the population of the village will first be greater than 3000
   (Total for Question 4 is 10 marks)
   TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS END

2. (a) Find the general solution of the recurrence relation $$u _ { n + 1 } = 3 u _ { n } + 2 ^ { n } \quad n \geqslant 1$$ (b) Find the particular solution of this recurrence relation for which \(u _ { 1 } = u _ { 2 }\)

4. Sarah takes out a mortgage of \(\pounds 155000\) to buy a house. Interest is added each month on the outstanding balance at a constant rate of \(r\) \% each month. Sarah makes fixed monthly repayments to reduce the amount owed. Each month, interest is added, and then her monthly repayment is used to reduce the outstanding amount owed. The recurrence relationship for the amount of the mortgage outstanding after \(n + 1\) months is modelled by $$u _ { n + 1 } = 1.0025 u _ { n } - x \quad n \geqslant 0$$ where \(\pounds u _ { n }\) is the amount of the mortgage outstanding after \(n\) months and \(\pounds x\) is the monthly repayment.

1. State the value of \(r\).
2. Solve the recurrence relation to find an expression for \(u _ { n }\) in terms of \(x\) and \(n\). Given that the mortgage will be paid off in exactly 30 years,
3. determine, to 2 decimal places, the least possible value of \(x\). \section*{(Total for Question 4 is 9 marks)} TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS
   END

4. A sequence \(\left\{ u _ { n } \right\}\), where \(n \geqslant 0\), satisfies the recurrence relation $$u _ { n + 1 } + 3 u _ { n } = n + k$$ where \(k\) is a non-zero constant.
Given that \(u _ { 0 } = 1\)

1. solve the recurrence relation, giving \(u _ { n }\) in terms of \(k\) and \(n\). Given that \(u _ { n }\) is a linear function of \(n\),
2. use your answer to part (a) to find the value of \(u _ { 100 }\) TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS END

4. Peter sets up a savings plan. He makes an initial deposit of \(\pounds D\) and then pays in \(\pounds M\) at the end of each month. The value of the savings plan, in pounds, is modelled by $$u _ { n + 1 } = 1.025 u _ { n } + 1800$$ where \(n \geqslant 0\) is an integer and \(u _ { n }\) is the total value of the savings plan, in pounds, after \(n\) years.

1. Calculate the value of \(M\) Given that the value of the savings plan after 1 year is \(\pounds 6925\)
2. solve the recurrence relation for \(u _ { n }\)
3. Determine the value of \(D\)
4. Hence determine, using algebra, the number of years it will take for the value of the savings plan to exceed \(\pounds 20000\)

5 An inspector has three factories, A, B, C, to check. He spends each day in one of the factories. He chooses the factory to visit on a particular day according to the following rules.

• If he is in A one day, then the next day he will never choose A but he is equally likely to choose B or C .
• If he is in B one day, then the next day he is equally likely to choose \(\mathrm { A } , \mathrm { B }\) or C .
• If he is in C one day, then the next day he will never choose A but he is equally likely to choose B or C .
  1. Write down the transition matrix, \(\mathbf { P }\).
  2. On Day 1 the inspector chooses A.
     (A) Find the probability that he will choose A on Day 4.
     (B) Find the probability that the factory he chooses on Day 7 is the same factory that he chose on Day 2.
  3. Find the equilibrium probabilities and explain what they mean.

The inspector is not satisfied with the number of times he visits A so he changes the rules as follows.

Still not satisfied, the inspector changes the rules as follows.

The new transition matrix is \(\mathbf { R }\).

On Day 15 he visits C . Find the first subsequent day for which the probability that he visits B is less than 0.1.

Show that in this situation there is an absorbing state, explaining what this means. \section*{END OF QUESTION PAPER}

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.

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   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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   \(B\)
   \(C\)
   \(D\)

   	1	2	3	4
   A
   B
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   D

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   2.

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   3. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea07472-f93b-4a7b-b362-89fb8c0af4a9-16_936_1317_255_376} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} 5.

3. 	\(R\)	\(S\)	\(T\)
   \(A\)
   \(B\)
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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}		□	□
   P

   8.

A student is testing a numerical method for finding an approximation for \(\pi\). The algorithm that the student is using is as follows. Line 10 \quad Input \(A\), \(B\), \(C\), \(D\), \(E\) Line 20 \quad Let \(A = A + 2\) Line 30 \quad Let \(B = -B\) Line 40 \quad Let \(C = \frac{B}{A}\) Line 50 \quad Let \(D = D + C\) Line 60 \quad Let \(E = (D - 3.14)^2\) Line 70 \quad If \(E < 0.05\) then go to Line 90 Line 80 \quad Go to Line 20 Line 90 \quad Print '\(\pi\) is approximately', \(D\) Line 100 \quad End Trace the algorithm in the case where the input values are $$A = 1, \quad B = 4, \quad C = 0, \quad D = 4, \quad E = 0$$ [6 marks]

In this question, the function INT(\(X\)) is the largest integer less than or equal to \(X\). For example, $$\text{INT}(3.6) = 3,$$ $$\text{INT}(3) = 3,$$ $$\text{INT}(-3.6) = -4.$$ Consider the following algorithm. \begin{align} \text{Step 1} \quad & \text{Input } B
\text{Step 2} \quad & \text{Input } N
\text{Step 3} \quad & \text{Calculate } F = N \div B
\text{Step 4} \quad & \text{Let } G = \text{INT}(F)
\text{Step 5} \quad & \text{Calculate } H = B \times G
\text{Step 6} \quad & \text{Calculate } C = N - H
\text{Step 7} \quad & \text{Output } C
\text{Step 8} \quad & \text{Replace } N \text{ by the value of } G
\text{Step 9} \quad & \text{If } N = 0 \text{ then stop, otherwise go back to Step 3} \end{align}

1. Apply the algorithm with the inputs \(B = 2\) and \(N = 5\). Record the values of \(F\), \(G\), \(H\), \(C\) and \(N\) each time Step 9 is reached. [5]
2. Explain what happens when the algorithm is applied with the inputs \(B = 2\) and \(N = -5\). [4]
3. Apply the algorithm with the inputs \(B = 10\) and \(N = 37\). Record the values of \(F\), \(G\), \(H\), \(C\) and \(N\) each time Step 9 is reached. What are the output values when \(B = 10\) and \(N\) is any positive integer? [4]

The function INT(\(C\)) gives the largest integer that is less than or equal to \(C\). For example: INT(4.8) = 4, INT(7) = 7, INT(0.8) = 0, INT(−0.8) = −1, INT(−2.4) = −3. Consider the following algorithm. Line 10 \quad Input \(A\) and \(B\) Line 20 \quad Calculate \(C = B \div A\) Line 30 \quad Let \(D =\) INT(\(C\)) Line 40 \quad Calculate \(E = A \times D\) Line 50 \quad Calculate \(F = B - E\) Line 60 \quad Output the value of \(F\) Line 70 \quad Replace \(B\) by the value of \(D\) Line 80 \quad If \(B = 0\) then stop, otherwise go back to line 20

1. Apply the algorithm using the inputs \(A = 10\) and \(B = 128\). Record the values of \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) every time they change. Record the output each time line 60 is reached. [4]
2. Show what happens when the input values are \(A = 10\) and \(B = -13\). [5]

A bag contains five pieces of paper labelled A, B, C, D and E. One piece is drawn at random from the bag. If the piece is labelled with a vowel (A or E) then the process stops. Otherwise the piece of paper is replaced, the bag is shaken, and the process is repeated. You are to simulate this process to estimate the mean number of draws needed to get a vowel.

1. Show how to use single digit random numbers to simulate the process efficiently. You need to describe exactly how your simulation will work. [3]
2. Use the random numbers in your answer book to run your simulation 5 times, recording your results. [2]
3. From your results compute an estimate of the mean number of draws needed to get a vowel. [2]
4. State how you could produce a more accurate estimate. [1]