Sequences and Series

7 Find \(\sum _ { r = 1 } ^ { n } 3 r ( r - 1 )\), expressing your answer in a fully factorised form.

1 Use the List of Formulae (MF10) to show that \(\sum _ { r = 1 } ^ { 13 } \left( 3 r ^ { 2 } - 5 r + 1 \right)\) and \(\sum _ { r = 0 } ^ { 9 } \left( r ^ { 3 } - 1 \right)\) have the same numerical value.

4．A sequence of positive integers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has \(r\) th term given by $$a _ { r } = 2 ^ { r } - 1$$ （a）Write down the first 6 terms of this sequence．
（b）Verify that \(a _ { r + 1 } = 2 a _ { r } + 1\)
（c）Find \(\sum _ { r = 1 } ^ { n } a _ { r }\)
（d）Show that \(\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }\)
（e）Hence show that \(1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)\)
（f）Show that \(\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }\)

5

1. Determine the general solution of the first-order recurrence relation \(V _ { n + 1 } = 2 V _ { n } + n\).
2. Given that \(V _ { 1 } = 8\), find the exact value of \(V _ { 20 }\).

3 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = k u _ { n } + 12$$ where \(k\) is a constant.
The first two terms of the sequence are given by $$u _ { 1 } = 16 \quad u _ { 2 } = 24$$

1. Show that \(k = 0.75\).
2. Find the value of \(u _ { 3 }\) and 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 \(L\).

3 A sequence is defined by the recurrence relation \(5 t _ { n + 1 } - 4 t _ { n } = 3 n ^ { 2 } + 28 n + 6\), for \(n \geqslant 0\), with \(t _ { 0 } = 7\).

1. Find an expression for \(t _ { n }\) in terms of \(n\). Another sequence is defined by \(\mathrm { v } _ { \mathrm { n } } = \frac { \mathrm { t } _ { \mathrm { n } } } { \mathrm { n } ^ { \mathrm { m } } }\), for \(n \geqslant 1\), where \(m\) is a constant.
2. In each of the following cases determine \(\lim _ { n \rightarrow \infty } \mathrm {~V} _ { n }\), if it exists, or show that the sequence is divergent.
   1. \(m = 3\)
   2. \(m = 2\)
   3. \(m = 1\)

1. Sam borrows \(\pounds 10000\) from a bank to pay for an extension to his house.

The bank charges \(5 \%\) annual interest on the portion of the loan yet to be repaid. Immediately after the interest has been added at the end of each year and before the start of the next year, Sam pays the bank a fixed amount, \(\pounds F\). Given that \(\pounds A _ { n }\) (where \(A _ { n } \geqslant 0\) ) is the amount owed at the start of year \(n\),

1. write down an expression for \(A _ { n + 1 }\) in terms of \(A _ { n }\) and \(F\),
2. prove, by induction that, for \(n \geqslant 1\) $$A _ { n } = ( 10000 - 20 F ) 1.05 ^ { n - 1 } + 20 F$$
3. Find the smallest value of \(F\) for which Sam can repay all of the loan by the start of year 16 .

7 In a conservation project, a batch of 100000 tadpoles which have just hatched from eggs is introduced into an environment which has no frog population. Previous research suggests that for every 1 million tadpoles hatched only 3550 will live to maturity at 12 weeks, when they become adult frogs. It is assumed that the steady decline in the population of tadpoles, from all causes, can be explained by a weekly death-rate factor, \(r\), which is constant across each week of this twelve-week period. Let \(\mathrm { T } _ { \mathrm { k } }\) denote the total number of tadpoles alive at the end of \(k\) weeks after the start of this project.

1. 1. Explain why a recurrence system for \(\mathrm { T } _ { \mathrm { k } }\) is given by \(T _ { 0 } = 100000\) and \(\mathrm { T } _ { \mathrm { k } + 1 } = ( 1 - \mathrm { r } ) \mathrm { T } _ { \mathrm { k } }\) for \(0 \leqslant k \leqslant 12\).
   2. Show that \(r = 0.375\), correct to 3 significant figures. The proportion of females within each batch of tadpoles is \(p\), where \(0 < p < 1\). In a simple model of the frog population the following assumptions are made.
      • The death rate factor for adult frogs is also \(r\) and is the same for males and females.
2. The frog population will survive provided there are at least thirty female frogs alive sixteen weeks after the start of this project.
3. 1. Find the smallest value of \(p\) for which the frog population will survive according to the model.
   2. Write down one assumption that has been made in order to obtain this result.
Each surviving female will then lay a batch of eggs from which 2500 tadpoles are hatched.
4. By considering the total number of tadpoles hatched, give one criticism of the assumption that the frog population will survive provided there are at least thirty female frogs alive sixteen weeks after the start of this project. \section*{END OF QUESTION PAPER}

5 Every day, a security firm transports a large sum of money from one bank to another. There are three possible routes \(A , B\) and \(C\). The route to be used is decided just before the journey begins, by a computer programmed as follows. On the first day, each of the three routes is equally likely to be used.
If route \(A\) was used on the previous day, route \(A\), \(B\) or \(C\) will be used, with probabilities \(0.1,0.4,0.5\) respectively.
If route \(B\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.7,0.2,0.1\) respectively.
If route \(C\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.1,0.6,0.3\) respectively. The situation is modelled as a Markov chain with three states.

1. Write down the transition matrix \(\mathbf { P }\).
2. Find the probability that route \(B\) is used on the 7th day.
3. Find the probability that the same route is used on the 7th and 8th days.
4. Find the probability that the route used on the 10th day is the same as that used on the 7th day.
5. Given that \(\mathbf { P } ^ { n } \rightarrow \mathbf { Q }\) as \(n \rightarrow \infty\), find the matrix \(\mathbf { Q }\) (give the elements to 4 decimal places). Interpret the probabilities which occur in the matrix \(\mathbf { Q }\). The computer program is now to be changed, so that the long-run probabilities for routes \(A , B\) and \(C\) will become \(0.4,0.2\) and 0.4 respectively. The transition probabilities after routes \(A\) and \(B\) remain the same as before.
6. Find the new transition probabilities after route \(C\).
7. A long time after the change of program, a day is chosen at random. Find the probability that the route used on that day is the same as on the previous day. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

2 A periodic sequence is defined by $$U _ { n } = ( - 1 ) ^ { n }$$ State the period of the sequence.
Circle your answer.

2 A periodic sequence is defined by $$U _ { n } = ( - 1 ) ^ { n }$$ State the period of the sequence.
Circle your answer.

1. A sequence of non-zero real numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by

$$a _ { n + 1 } = p + \frac { q } { a _ { n } } \quad n \in \mathbb { N }$$ where \(p\) and \(q\) are real numbers with \(q \neq 0\)
It is known that

• one of the terms of this sequence is a
• the sequence is periodic
  1. Determine an equation for \(q\), in terms of \(p\) and \(a\), such that the sequence is constant (of period/order one).
  2. Determine the value of \(p\) that is necessary for the sequence to be of period/order 2.
  3. Give an example of a sequence that satisfies the condition in part (b), but is not of period/order 2.
  4. Determine an equation for \(q\), in terms of \(p\) only, such that the sequence has period/order 4.

1 Evaluate \(\sum _ { r = 101 } ^ { 250 } r ^ { 3 }\).

1 Evaluate \(\sum _ { r = 101 } ^ { 250 } r ^ { 3 }\).

4 In this question you must show detailed reasoning.
The series \(S\) is defined as being the sum of the squares of all positive odd integers from \(1 ^ { 2 }\) to \(779 ^ { 2 }\). Determine the value of \(S\).

\(\mathbf { 5 }\) & 7 & 6 & \(\infty\) & 10 & 14 & 8
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4

1. Solve the second-order recurrence relation \(T _ { n + 2 } + 2 T _ { n } = - 87\) given that \(T _ { 0 } = - 27\) and \(T _ { 1 } = 27\).
2. Determine the value of \(T _ { 20 }\).

3
Given \(u _ { 1 } = 1\), determine which one of the formulae below defines an increasing sequence for \(n \geq 1\) Circle your answer.
[0pt] [1 mark]
\(u _ { n + 1 } = 1 + \frac { 1 } { u _ { n } } \quad u _ { n } = 2 - 0.9 ^ { n - 1 } \quad u _ { n + 1 } = - 1 + 0.5 u _ { n } \quad u _ { n } = 0.9 ^ { n - 1 }\)

1 The sequence \(\left\{ u _ { n } \right\}\) is defined by \(u _ { 1 } = 2\) and \(u _ { n + 1 } = \frac { 12 } { 1 + u _ { n } }\) for \(n \geq 1\).
Given that the sequence converges, with limit \(\alpha\), determine the value of \(\alpha\).

1

1. Use the list of formulae (MF19) to find \(\sum _ { r = 1 } ^ { n } r ( r + 2 )\) in terms of \(n\), simplifying your answer.
2. Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions and hence find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { \mathrm { r } ( \mathrm { r } + 2 ) }\) in terms of \(n\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 2 ) }\).

4. A sequence is defined by \(u _ { 1 } = 3 , u _ { n + 1 } = u _ { n } ^ { r }\) for \(n \geq 1\).
a) In the case where \(r = \frac { 6 } { 5 }\) find the smallest value of \(n\) such that \(u _ { n } > 10 ^ { 50 }\). A convergent sequence is defined by \(v _ { 1 } = u _ { 1 } , v _ { n + 1 } = u _ { n + 1 } v _ { n }\) for \(n \geq 1\).
b) Given that the limit of this sequence is greater than 100 , find the range of possible values of \(r\), giving your answer in exact form.
c) Evaluate the infinite product: $$2 \times \sqrt [ 3 ] { 4 } \times \sqrt [ 3 ] { \sqrt [ 3 ] { 16 } } \times \sqrt [ 3 ] { \sqrt [ 3 ] { \sqrt [ 3 ] { 256 } } } \cdots$$ [Question 4 - Continued]
[0pt] [Question 4 - Continued]
[0pt] [Question 4 - Continued]

4 Given that \(\sum _ { r = 1 } ^ { n } \left( a r ^ { 3 } + b r \right) \equiv n ( n - 1 ) ( n + 1 ) ( n + 2 )\), find the values of the constants \(a\) and \(b\).

6 The Fibonacci sequence \(\left\{ F _ { n } \right\}\) is defined by \(F _ { 0 } = 0 , F _ { 1 } = 1\) and \(F _ { n } = F _ { n - 1 } + F _ { n - 2 }\) for all \(n \geqslant 2\).

1. Show that \(F _ { n + 5 } = 5 F _ { n + 1 } + 3 F _ { n }\)
2. Prove that \(F _ { n }\) is a multiple of 5 when \(n\) is a multiple of 5 .

4 The sequence \(u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots\) satisfies the recurrence relation \(u _ { n + 2 } - 3 u _ { n + 1 } - 10 u _ { n } = 24 n - 10\).

1. Determine the general solution of the recurrence relation.
2. Hence determine the particular solution of the recurrence relation for which \(u _ { 0 } = 6\) and \(u _ { 1 } = 10\).
3. Show, by direct calculation, that your solution in part (b) gives the correct value for \(u _ { 2 }\). The sequence \(v _ { 0 } , v _ { 1 } , v _ { 2 } , \ldots\) is defined by \(v _ { n } = \frac { u _ { n } } { p ^ { n } }\) for some constant \(p\), where \(u _ { n }\) denotes the
   particular solution found in part (b). particular solution found in part (b). You are given that \(\mathrm { v } _ { \mathrm { n } }\) converges to a finite non-zero limit, \(q\), as \(n \rightarrow \infty\).
4. Determine \(p\) and \(q\).

1. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by

$$\begin{aligned} u _ { n + 1 } & = b - a u _ { n } \\ u _ { 1 } & = 3 \end{aligned}$$ where \(a\) and \(b\) are constants.

1. Find, in terms of \(a\) and \(b\),
   1. \(u _ { 2 }\)
   2. \(u _ { 3 }\) Given
      • \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 153\)
2. \(b = a + 9\)
3. show that
$$a ^ { 2 } - 5 a - 66 = 0$$
4. Hence find the larger possible value of \(u _ { 2 }\)

7

1. Show that \(\sum _ { r = 1 } ^ { n } z ^ { 2 r } = \frac { z ^ { 2 n + 1 } - z } { z - z ^ { - 1 } }\), for \(z \neq 0,1 , - 1\).
2. By letting \(z = \cos \theta + i \sin \theta\), show that, if \(\sin \theta \neq 0\), $$1 + 2 \sum _ { r = 1 } ^ { n } \cos ( 2 r \theta ) = \frac { \sin ( 2 n + 1 ) \theta } { \sin \theta }$$

8 A television quiz show takes place every day. On day 1 the prize money is \(\\) 1000$. If this is not won the prize money is increased for day 2 . The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money. Model 1: Increase the prize money by \(\\) 1000$ each day.
Model 2: Increase the prize money by \(10 \%\) each day.
On each day that the prize money is not won the television company makes a donation to charity. The amount donated is \(5 \%\) of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity

1. if Model 1 is used,
2. if Model 2 is used.