First-Order Linear Recurrence Relations

4 The sequence \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is defined by \(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = 2 \mathrm { u } _ { \mathrm { n } } + \mathrm { n } ^ { 2 }\) for \(\mathrm { n } \geqslant 1\).
Determine \(u _ { n }\) as a function of \(n\).

6 The sequence \(\left\{ u _ { n } \right\}\) is such that \(u _ { 1 } = 7 , u _ { 2 } = 37 , u _ { 3 } = 337 , u _ { 4 } = 3337 , \ldots\).

1. Write down a first-order recurrence system for \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\).
2. By solving the recurrence system of part (a), show that \(\mathrm { u } _ { \mathrm { n } } = \frac { 1 } { 3 } \left( 10 ^ { \mathrm { n } } + 11 \right)\).
3. Prove that \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) contains infinitely many terms which are multiples of 37 .

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 \(V _ { 20 }\).

6 In a national park, the number of adults of a given species is carefully monitored and controlled. The number of adults, \(n\) months after the start of this project, is \(A _ { n }\). Initially, there are 1000 adults. It is predicted that this number will have declined to 960 after one month. The first model for the number of adults is that, from one month to the next, a fixed proportion of adults is lost. In order to maintain a fixed number of adults, the park managers "top up" the numbers by adding a constant number of adults from other parks at the end of each month.

1. Use this model to express the number of adults as a first-order recurrence system. Instead, it is found that, the proportion of adults lost each month is double the predicted amount, with no change being made to the constant number of adults added each month.
2. 1. Show that the revised recurrence system for \(A _ { n }\) is \(A _ { 0 } = 1000 , A _ { n + 1 } = 0.92 A _ { n } + 40\). [1]
   2. Solve this revised recurrence system.
   3. Describe the long-term behaviour of the sequence \(\left\{ A _ { n } \right\}\) in this case. A more refined model for the number of adults uses the second-order recurrence system \(\mathrm { A } _ { \mathrm { n } + 1 } = 0.9 \mathrm {~A} _ { \mathrm { n } } - 0.1 \mathrm {~A} _ { \mathrm { n } - 1 } + 50\), for \(n \geqslant 1\), with \(A _ { 0 } = 1000\) and \(A _ { 1 } = 920\).
3. 1. Determine the long-term behaviour of the sequence \(\left\{ A _ { n } \right\}\) for this more refined model.
   2. A criticism of this more refined model is that it does not take account of the fact that the number of adults must be an integer at all times. State a modified form of the second-order recurrence relation for this more refined model that will satisfy this requirement.

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 \(L\).

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 \(L\).

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 \(u _ { 3 }\).

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. \(m = 1\)

2 A sequence is defined by the recurrence relation \(4 \mathrm { t } _ { \mathrm { n } + 1 } - \mathrm { t } _ { \mathrm { n } } = 15 \mathrm { n } + 17\) for \(\mathrm { n } \geqslant 1\), with \(t _ { 1 } = 2\).

1. Solve the recurrence relation to find the particular solution for \(\mathrm { t } _ { \mathrm { n } }\). Another sequence is defined by the recurrence relation \(( n + 1 ) u _ { n + 1 } - u _ { n } ^ { 2 } = 2 n - \frac { 1 } { n ^ { 2 } }\) for \(n \geqslant 1\), with \(u _ { 1 } = 2\).
2. 1. Explain why the recurrence relation for \(\mathrm { u } _ { \mathrm { n } }\) cannot be solved using standard techniques for non-homogeneous first order recurrence relations.
   2. Verify that the particular solution to this recurrence relation is given by \(u _ { n } = a n + \frac { b } { n }\) where \(a\) and \(b\) are constants whose values are to be determined. A third sequence is defined by \(\mathrm { v } _ { \mathrm { n } } = \frac { \mathrm { t } _ { \mathrm { n } } } { \mathrm { u } _ { \mathrm { n } } }\) for \(n \geqslant 1\).
3. Determine \(\lim _ { n \rightarrow \infty } \mathrm { v } _ { \mathrm { n } }\).

2 A sequence is defined by the recurrence relation \(t _ { n + 1 } = \frac { t _ { n } } { n + 3 }\) for \(n \geqslant 1\), with \(t _ { 1 } = 8\).
Verify that the particular solution to the recurrence relation is given by \(t _ { n } = \frac { a } { ( n + b ) ! }\) where \(a\) and \(b\) are constants whose values are to be determined.

7 Irrational numbers can be modelled by sequences \(\left\{ u _ { n } \right\}\) of rational numbers of the form $$u _ { 0 } = 1 \text { and } u _ { n + 1 } = a + \frac { 1 } { b + u _ { n } } \text { for } n \geqslant 0 \text {, }$$ where \(a\) and \(b\) are non-negative integer constants.

1. (a) The constants \(a = 1\) and \(b = 0\) produce the irrational number \(\omega\). State the value of \(\omega\) correct to six decimal places.
   (b) By setting \(u _ { n + 1 }\) and \(u _ { n }\) equal to \(\omega\), determine the exact value of \(\omega\).
2. Use the method of part (i) (b) to find the exact value of the irrational number produced by taking \(a = 0\) and \(b = 2\).
3. Find positive integers \(a\) and \(b\) which would produce the irrational number \(2 + \sqrt { 10 }\). \section*{END OF QUESTION PAPER}

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.