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 $$a _ { r } = 2 ^ { r } - 1$$

1. Write down the first 6 terms of this sequence.
2. Verify that \(a _ { r + 1 } = 2 a _ { r } + 1\)
3. Find \(\sum _ { r = 1 } ^ { n } a _ { r }\)
4. Show that \(\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }\)
5. Hence show that \(1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)\)
6. Show that \(\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }\)

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.

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 \(V _ { 20 }\).

A sequence \(a_1, a_2, a_3, \ldots\) is defined by $$a_1 = 3,$$ $$a_{n+1} = 3a_n - 5, \quad n \geq 1.$$

1. Find the value \(a_2\) and the value of \(a_3\). [2]
2. Calculate the value of \(\sum_{r=1}^5 a_r\). [3]

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. \(m = 1\)

Initially the number of fish in a lake is 500 000. The population is then modelled by the recurrence relation $$u_{n+1} = 1.05u_n - d, \quad u_0 = 500000.$$ In this relation \(u_n\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year. Given that \(d = 15000\),

1. calculate \(u_1\), \(u_2\) and \(u_3\) and comment briefly on your results. [3]

Given that \(d = 100000\),

1. show that the population of fish dies out during the sixth year. [3]
2. Find the value of \(d\) which would leave the population each year unchanged. [2]

Initially the number of fish in a lake is 500 000. The population is then modelled by the recurrence relation $$u_{n+1} = 1.05u_n - d, \quad u_0 = 500000.$$ In this relation \(u_n\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year. Given that \(d = 15000\),

1. calculate \(u_1\), \(u_2\) and \(u_3\) and comment briefly on your results. [3]

Given that \(d = 100000\),

1. show that the population of fish dies out during the sixth year. [3]
2. Find the value of \(d\) which would leave the population each year unchanged. [2]

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.

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

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 \(T _ { 20 }\).

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 \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 2 ) }\).

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 .

For each of the following sequences, find the first 5 terms, \(u _ { 1 }\) to \(u _ { 5 }\). Describe the behaviour of each sequence. a) \(\quad u _ { n } = \sin \left( \frac { n \pi } { 2 } \right)\) b) \(u _ { 6 } = 33 , u _ { n } = 2 u _ { n - 1 } - 1\)

1 It is given that \(\sum _ { r = 1 } ^ { n } u _ { r } = n ^ { 2 } ( 2 n + 3 )\), where \(n\) is a positive integer.

1. Find \(\sum _ { r = n + 1 } ^ { 2 n } u _ { r }\).
2. Find \(u _ { r }\).

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 \(b\).

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]

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.

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 \(\sin \theta \neq 0\), $$1 + 2 \sum _ { r = 1 } ^ { n } \cos ( 2 r \theta ) = \frac { \sin ( 2 n + 1 ) \theta } { \sin \theta }$$

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 \(u _ { 2 }\)

2

1. Determine the general solution of the recurrence relation \(2 u _ { n + 2 } - 7 u _ { n + 1 } + 3 u _ { n } = 0\).
2. Using your answer to part (a), determine the general solution of the recurrence relation \(2 u _ { n + 2 } - 7 u _ { n + 1 } + 3 u _ { n } = 20 n ^ { 2 } + 60 n\). In the rest of this question the sequence \(u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots\) satisfies the recurrence relation in part (b). You are given that \(u _ { 0 } = - 9\) and \(u _ { 1 } = - 12\).
3. Determine the particular solution for \(\mathrm { u } _ { \mathrm { n } }\). You are given that, as \(n\) increases, once the values of \(u _ { n }\) start to increase, then from that point onwards the sequence is an increasing sequence.
4. Use your answer to part (c) to determine, by direct calculation, the least value taken by terms in the sequence. You should show any values that you rely on in your argument.