Iterative formula convergence

3 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 3 x _ { n } } { 4 } + \frac { 15 } { x _ { n } ^ { 3 } }$$ with initial value \(x _ { 1 } = 3\), converges to \(\alpha\).

1. Use this iterative formula to find \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
2. State an equation satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).

6 A sequence is defined by $$a _ { n + 1 } = 2 a _ { n } + 3 a _ { n - 1 } \quad \text { with } a _ { 1 } = 1 \text { and } a _ { 2 } = 1 .$$ Using the method on page 5, show that the value to which the ratio of successive terms converges is 3 .
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6 When \(10 ^ { 6 }\) of a certain type of bacteria are detected in a blood sample of an infected animal, a course of treatment is started. The long-term aim of the treatment is to reduce the number of bacteria in such a sample to under 10000 . At this level the animal's immune system can fight the infection for itself. Once treatment has started, if the number of bacteria in a sample is 10000 or more, then treatment either continues or restarts. The model suggested to predict the progress of the course of treatment is based on the recurrence system \(P _ { n + 1 } = \frac { 2 P _ { n } } { n + 1 } + \frac { n } { P _ { n } }\) for \(n \geqslant 0\), with \(P _ { 0 } = 1000\), where \(P _ { n }\) denotes the number of bacteria (in thousands) present in the animal's body \(n\) days after the treatment was started. The table below shows the values of \(P _ { n }\), for certain chosen values of \(n\). Each value has been given correct to 2 decimal places (where appropriate).

1. Find the value of \(P _ { 6 }\) correct to 2 decimal places.
2. Using the given values for \(P _ { 0 }\) to \(P _ { 9 }\), and assuming that the model is valid,
   1. describe the effects of this course of treatment during the first 9 days,
   2. state the number of days after treatment is started when the animal's own immune system is expected to be able to fight the infection for itself.
3. 1. Using information from the above table, suggest a function f such that, for \(n > 10 , \mathrm { f } ( n )\) is a suitable approximation for \(P _ { n }\).
   2. Use your suggested function to estimate the number of days after treatment is started when the animal may once again require medical intervention in order to help fight off this bacterial infection.
   3. Using information from the above table and the recurrence relation, verify or correct the estimate which you found in part (c)(ii).
4. One criticism of the system \(P _ { n + 1 } = \frac { 2 P _ { n } } { n + 1 } + \frac { n } { P _ { n } }\), with \(P _ { 0 } = 1000\), is that it gives non-integer
   values of \(P\). values of \(P _ { n }\). Suggest a modification that would correct this issue.

5 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where \(p\) and \(q\) are constants.
The first three terms of the sequence are given by $$u _ { 1 } = 200 \quad u _ { 2 } = 150 \quad u _ { 3 } = 120$$

1. Show that \(p = 0.6\) and find the value of \(q\).
2. Find the value of \(u _ { 4 }\).
3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\).

6 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where \(p\) and \(q\) are constants.
The first three terms of the sequence are given by $$u _ { 1 } = - 8 \quad u _ { 2 } = 8 \quad u _ { 3 } = 4$$

1. Show that \(q = 6\) and find the value of \(p\).
2. Find the value of \(u _ { 4 }\).
3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\).
   1. Write down an equation for \(L\).
   2. Hence find the value of \(L\).

2 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = 6 + \frac { 2 } { 5 } u _ { n }$$ The first term of the sequence is given by \(u _ { 1 } = 2\).

1. Find the value of \(u _ { 2 }\) and the value of \(u _ { 3 }\).
2. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\).

7 The \(n\)th term of a sequence is \(u _ { n }\). The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where \(p\) and \(q\) are constants.
The first two terms of the sequence are given by \(u _ { 1 } = 60\) and \(u _ { 2 } = 48\).
The limit of \(u _ { n }\) as \(n\) tends to infinity is 12 .

1. Show that \(p = \frac { 3 } { 4 }\) and find the value of \(q\).
2. Find the value of \(u _ { 3 }\).

5 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by \(u _ { n + 1 } = p u _ { n } + q\), where \(p\) and \(q\) are constants.
The second term of the sequence is 160 . The third term of the sequence is 132 .
The limit of \(u _ { n }\) as \(n\) tends to infinity is 20 .

1. Find the value of \(p\) and the value of \(q\).
2. Hence find the value of the first term of the sequence.

1. A student takes out a loan for \(\pounds 1000\) from a bank.

The bank charges \(0.5 \%\) monthly interest on the amount of the loan yet to be repaid.
At the end of each month

• the interest is added to the loan
• the student then repays \(\pounds 50\)

Let \(U _ { n }\) be the amount of money owed \(n\) months after the loan was taken out.
The amount of money owed by the student is modelled by the recurrence relation $$U _ { n } = 1.005 U _ { n - 1 } - A \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$ where \(A\) is a constant.

1. 1. State the value of the constant \(A\).
   2. Explain, in the context of the problem, the value 1.005 Using the value of \(A\) found in part (a)(i),
2. solve the recurrence relation $$U _ { n } = 1.005 U _ { n - 1 } - A \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$
3. Hence determine, according to the model, the number of months it will take to completely repay the loan.

1. A population of deer on a large estate is assumed to increase by \(10 \%\) during each year due to natural causes.

The population is controlled by removing a constant number, \(Q\), of the deer from the estate at the end of each year. At the start of the first year there are 5000 deer on the estate.
Let \(P _ { n }\) be the population of deer at the end of year \(n\).

1. Explain, in the context of the problem, the reason that the deer population is modelled by the recurrence relation $$P _ { n } = 1.1 P _ { n - 1 } - Q , \quad P _ { 0 } = 5000 , \quad n \in \mathbb { Z } ^ { + }$$
2. Prove by induction that \(P _ { n } = ( 1.1 ) ^ { n } ( 5000 - 10 Q ) + 10 Q , \quad n \geqslant 0\)
3. Explain how the long term behaviour of this population varies for different values of \(Q\).

7. Martina decides to open a bank account to help her to save for a holiday. Each month she puts \(\pounds \mathrm { k }\) into the account and allows herself to spend one quarter of what was in the account at the end of the previous month. Let \(u _ { n }\) (where \(\mathrm { n } \geqslant 1\) ) represent the amount in the account at the end of month n .
Martina has \(\pounds \mathrm {~K}\) in the account 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 and k At the end of the 8th month, Martina needs to have at least \(\pounds 1750\) in the account to pay for her holiday.
2. Determine, to the nearest penny, the minimum amount of money that Martina should put into the account each month.

11 The \(n\)th term of a sequence is \(u _ { n }\) The sequence is defined by $$u _ { n + 1 } = p u _ { n } + 70$$ where \(u _ { 1 } = 400\) and \(p\) is a constant.
11

1. Find an expression, in terms of \(p\), for \(u _ { 2 }\) 11
2. It is given that \(u _ { 3 } = 382\) 11
3. 1. Show that \(p\) satisfies the equation $$200 p ^ { 2 } + 35 p - 156 = 0$$ 11
4. (ii) It is given that the sequence is a decreasing sequence. Find the value of \(u _ { 4 }\) and the value of \(u _ { 5 }\) 11
5. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\) 11
6. 1. Write down an equation for \(L\) 11
7. (ii) Find the value of \(L\)

7 In a long-running biochemical experiment, an initial amount of 1200 mg of an enzyme is placed into a mixture. The model for the amount of enzyme present in the mixture suggests that, at the end of each hour, one-eighth of the amount of enzyme that was present at the start of that hour is used up due to chemical reactions within the mixture. To compensate for this, at the end of each six-hour period of time, a further 500 mg of the enzyme is added to the mixture.

1. Let \(n\) be the number of six-hour periods that have elapsed since the experiment began. Explain how the amount of enzyme, \(\mathrm { E } _ { \mathrm { n } } \mathrm { mg }\), in the mixture is given by the recurrence system \(E _ { 0 } = 1200\) and \(E _ { n + 1 } = \left( \frac { 7 } { 8 } \right) ^ { 6 } E _ { n } + 500\) for \(n \geqslant 0\).
2. Solve the recurrence system given in part (a) to obtain an exact expression for \(\mathrm { E } _ { \mathrm { n } }\) in terms of \(n\).
3. Hence determine, in the long term, the amount of enzyme in the mixture. Give your answer correct to \(\mathbf { 3 }\) significant figures.
4. In this question you must show detailed reasoning. The long-running experiment is then repeated. This time a new requirement is added that the amount of enzyme present in the mixture must always be at least 500 mg . Show that the new requirement ceases to be satisfied before 12 hours have elapsed. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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8 A sequence \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is defined by the recurrence system \(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = \mathrm { a } - \frac { \mathrm { a } ^ { 2 } } { 2 \mathrm { u } _ { \mathrm { n } } }\) for \(n \geqslant 1\), where \(a\) is a positive constant.
Determine with justification the behaviour of the sequence for all possible values of \(a\). \section*{END OF QUESTION PAPER}