1.04e Sequences: nth term and recurrence relations

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 \(u _ { 3 }\).

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 \(u _ { 1 } = 96\) and \(u _ { 2 } = 72\).
The limit of \(u _ { n }\) as \(n\) tends to infinity is 24 .

1. Show that \(p = \frac { 2 } { 3 }\).
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.

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.

7 Sam decided to go on a high-protein diet. Sam's mass in \(\mathrm { kg } , M\), after \(t\) days of following the diet is recorded in Fig. 7.1. \begin{table}[h] \end{table} A difference table for the data is shown in Fig. 7.2. \begin{table}[h] \end{table}

1. Complete the copy of the difference table in the Printed Answer Booklet. Sam's doctor uses these data to construct a cubic interpolating polynomial to model Sam's mass at time \(t\) days after starting the diet.
2. Find the model in the form \(\mathrm { M } = \mathrm { at } ^ { 3 } + \mathrm { bt } ^ { 2 } + \mathrm { ct } + \mathrm { d }\), where \(a , b , c\) and \(d\) are constants to be determined. Subsequently it is found that when \(\mathrm { t } = 40 , \mathrm { M } = 78.7\) and when \(\mathrm { t } = 50 , \mathrm { M } = 80.05\).
3. Determine whether the model is a good fit for these data.
4. By completing the extended copy of Fig. 7.2 in the Printed Answer Booklet, explain why a quartic model may be more appropriate for the data.
5. Refine the doctor's model to include a quartic term.
6. Explain whether the new model for Sam's mass is likely to be appropriate over a longer period of time.

4 Between 1946 and 2012 the mean monthly maximum temperature of the water surface of a lake in northern England has been recorded by environmental scientists. Some of the data are shown in Table 4.1. \begin{table}[h] \end{table} Table 4.2 shows a difference table for the data. \begin{table}[h] \end{table}

1. Complete the copy of the difference table in the Printed Answer Booklet.
2. Explain why a quadratic model may be appropriate for these data.
3. Use Newton's forward difference interpolation formula to construct an interpolating polynomial of degree 2 for these data. This polynomial is used to model the relationship between \(T\) and \(t\). Between 1946 and 2012 the mean monthly maximum temperature of the water surface of the lake was recorded as \(8.9 ^ { \circ } \mathrm { C }\) for October and \(7.5 ^ { \circ } \mathrm { C }\) for November.
4. Determine whether the model is a good fit for the temperatures recorded in October and November. A scientist recorded the mean monthly maximum temperature of the water surface of the lake in 2022. Some of the data are shown in Table 4.3. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 4.3}

   Month	May	June	July	August	September
   \(t =\) Time in months	0	1	2	3	4
   \(T =\) Mean temperature in \({ } ^ { \circ } \mathrm { C }\)	10.3	14.7	16.9	16.9	14.8

   \end{table}
5. Adapt the polynomial found in part (c) so that it can be used to model the relationship between \(T\) and \(t\) for the data in Table 4.3.

1 Three sequences, \(\mathrm { a } _ { \mathrm { n } } , \mathrm { b } _ { \mathrm { n } }\) and \(\mathrm { c } _ { \mathrm { n } }\), are defined for \(n \geqslant 1\) by the following recurrence relations. $$\begin{aligned} & \left( a _ { n + 1 } - 2 \right) \left( 2 - a _ { n } \right) = 3 \text { with } a _ { 1 } = 3 \\ & b _ { n + 1 } = - \frac { 1 } { 2 } b _ { n } + 3 \text { with } b _ { 1 } = 1.5 \\ & c _ { n + 1 } - \frac { c _ { n } ^ { 2 } } { n } = 1 \text { with } c _ { 1 } = 2.5 \end{aligned}$$ The output from a spreadsheet which presents the first 10 terms of \(a _ { n } , b _ { n }\) and \(c _ { n }\), is shown below. Without attempting to solve any recurrence relations, describe the apparent behaviour, including as \(n \rightarrow \infty\), of

• \(a _ { n }\)
• \(\mathrm { b } _ { \mathrm { n } }\)
• \(\mathrm { C } _ { \mathrm { n } }\)

1. A person takes a course of a particular vitamin.

Before the course there was none of the vitamin in the person's body.
During the course, vitamin tablets are taken at the same time each day.
Initially two tablets are taken and on each following day only one tablet is taken.
Each tablet contains 10 mg of the vitamin.
Between doses the amount of the vitamin in the person's body decreases naturally by 60\% Let \(u _ { n } \mathrm { mg }\) be the amount of the vitamin in the person's body immediately after a tablet is taken, \(n\) days after the initial two tablets were taken.

1. Explain why \(u _ { n }\) satisfies the recurrence relation $$u _ { 0 } = 20 \quad u _ { n + 1 } = 0.4 u _ { n } + 10$$ The general solution to this recurrence relation has the form \(u _ { n } = a ( 0.4 ) ^ { n } + b\)
2. Determine the value of \(a\) and the value of \(b\). The course is only effective if the amount of the vitamin in the person's body remains above 6 mg at all times throughout the course.
3. Determine whether this course of the vitamin will be effective for this person, giving a reason for your answer.

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.

5. \begin{figure}[h] \end{figure} Figure 1 shows the first three stages of a pattern that is created by a recursive process.
The process starts with a square and proceeds as follows

• each square is replaced by 5 smaller squares each \(\frac { 1 } { 9 }\) th the size of the square being replaced
• the 5 smaller squares are the ones in each corner and the one in the centre
• once each of the squares has been replaced, the square immediately to the right and above the centre square of the pattern is then removed

Let \(u _ { n }\) be the number of squares in the pattern in stage \(n\), where stage 1 is the original square.

1. Explain why \(u _ { n }\) satisfies the recurrence system $$u _ { 1 } = 1 \quad u _ { n + 1 } = 5 u _ { n } - 1 \quad ( n = 1,2,3 , \ldots )$$
2. Solve this recurrence system. Given that the initial square has area 25
3. determine the total area of all the squares in stage 8 of the pattern, giving your answer to 2 significant figures.

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 \(Q\).

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. 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. 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. 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)

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 \(\pounds 20000\)

2. In two-dimensional space, lines divide a plane into a number of different regions. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} It is known that:

• One line divides a plane into 2 regions, as shown in Figure 1
• Two lines divide a plane into a maximum of 4 regions, as shown in Figure 2
• Three lines divide a plane into a maximum of 7 regions, as shown in Figure 3
• Four lines divide a plane into a maximum of 11 regions, as shown in Figure 4

1. The number of visits to a website, in any particular month, is modelled as the number of visits received in the previous month plus \(k\) times the number of visits received in the month before that, where \(k\) is a positive constant.

Given that \(V _ { n }\) is the number of visits to the website in month \(n\),

1. write down a general recurrence relation for \(V _ { n + 2 }\) in terms of \(V _ { n + 1 } , V _ { n }\) and \(k\). For a particular website you are given that
   • \(k = 0.24\)
   • In month 1 , there were 65 visits to the website.
   • In month 2 , there were 71 visits to the website.
   • Show that
   $$V _ { n } = 50 ( 1.2 ) ^ { n } - 25 ( - 0.2 ) ^ { n }$$ This model predicts that the number of visits to this website will exceed one million for the first time in month \(N\).
2. Find the value of \(N\).

1. Solve the recurrence system

$$\begin{gathered} u _ { 1 } = 1 \quad u _ { 2 } = 4 \\ 9 u _ { n + 2 } - 12 u _ { n + 1 } + 4 u _ { n } = 3 n \end{gathered}$$

1. In a model for the number of subscribers to a new social media channel it is assumed that

• each week \(20 \%\) of the subscribers at the start of the week cancel their subscriptions
• between the start and end of week \(n\) the channel gains \(20 n\) new subscribers

Given that at the end of week 1 there were 25 subscribers,

1. explain why the number of subscribers at the end of week \(n , U _ { n }\), is modelled by the recurrence relation $$U _ { 1 } = 25 \quad U _ { n + 1 } = 0.8 U _ { n } + 20 ( n + 1 ) \quad n = 1,2,3 , \ldots$$
2. Prove by induction that for \(n \geqslant 1\) $$U _ { n } = 325 \left( \frac { 4 } { 5 } \right) ^ { n - 1 } + 100 n - 400$$ Given that 6 months after starting the channel there were approximately 1800 subscribers,
3. evaluate the model in the light of this information.

2 A sequence \(\left\{ u _ { n } \right\}\) is given by \(u _ { n + 1 } = 4 u _ { n } + 1\) for \(n \geqslant 1\) and \(u _ { 1 } = 3\).

1. Find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
2. Solve the recurrence system (*).
3. 1. Prove by induction that each term of the sequence can be written in the form \(( 10 m + 3 )\) where \(m\) is an integer.
   2. Show that no term of the sequence is a square number.

2. The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is given by: $$u _ { n + 1 } = \left( u _ { n } - 3 \right) ^ { 2 } , \quad u _ { 1 } = 1$$

1. Find \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
2. Write down the value of \(u _ { 20 }\).

7 Consecutive terms of a sequence are related by $$u _ { n + 1 } = 3 - \left( u _ { n } \right) ^ { 2 }$$ 7

1. In the case that \(u _ { 1 } = 2\) 7
   1. (i) Find \(u _ { 3 }\) 7
   2. (ii) Find \(u _ { 50 }\) 7
   3. State a different value for \(u _ { 1 }\) which gives the same value for \(u _ { 50 }\) as found in part (a)(ii).