Second-Order Homogeneous Recurrence Relations

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 \(T _ { 20 }\).

4

1. Solve the recurrence relation \(u _ { n + 2 } = 4 u _ { n + 1 } - 4 u _ { n }\) for \(n \geq 0\), given that \(u _ { 0 } = 1\) and \(u _ { 1 } = 1\).
2. Show that each term of the sequence \(\left\{ u _ { n } \right\}\) is an integer.

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

1. Find an exact real expression for \(u _ { n }\) in terms of \(n\) and \(\theta\), where \(\tan \theta = \frac { 1 } { 2 }\). A sequence is defined by \(v _ { n } = a ^ { \frac { 1 } { 2 } n } u _ { n }\) for \(n \geqslant 0\), where \(a\) is a positive constant.
2. In each of the following cases, describe the behaviour of \(v _ { n }\) as \(n \rightarrow \infty\).
   • \(a = 0.1\)
   • \(a = 0.2\)
   • \(a = 1\)

5. An increasing sequence \(\left\{ u _ { n } \right\}\) for \(n \in \mathbb { N }\) is such that the difference between the \(n\)th term of \(\left\{ u _ { n } \right\}\) and the mean of the previous two terms of \(\left\{ u _ { n } \right\}\) is always 6

1. Show that, for \(n \geqslant 3\) $$2 u _ { n } - u _ { n - 1 } - u _ { n - 2 } = 12$$ Given that \(u _ { 1 } = 2\) and \(u _ { 2 } = 8\)
2. find the solution of this second order recurrence relation to obtain an expression for \(u _ { n }\) in terms of \(n\).
3. Show that as \(n \rightarrow \infty , u _ { n } \rightarrow k n\) where \(k\) is a constant to be determined. You must give reasons for your answer.

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1. (a) Solve the recurrence relation $$X _ { n + 2 } = 1.3 X _ { n + 1 } + 0.3 X _ { n } \text { for } n \geqslant 0$$ given that \(X _ { 0 } = 12\) and \(X _ { 1 } = 1\).
   (b) Show that the sequence \(\left\{ X _ { n } \right\}\) approaches a geometric sequence as \(n\) increases. The recurrence relation in part (i) models the projected annual profit for an investment company, so that \(X _ { n }\) represents the profit (in \(\pounds\) ) at the end of year \(n\).
2. (a) Determine the number of years taken for the projected profit to exceed one million pounds.
   (b) Compare your answer to part (ii)(a) with the corresponding figure given by the geometric sequence of part (i)(b).
3. (a) In a modified model, any non-integer values obtained are rounded down to the nearest integer at each step of the process. Write down the recurrence relation for this model.
   (b) Write down the recurrence relation for the model in which any non-integer values obtained are rounded up to the nearest integer at each step of the process.
   (c) Describe a situation that might arise in the implementation of part (iii)(b) that would result in an incorrect value for the next \(X _ { n }\) in the process.

1. The complementary function for the second order recurrence relation

$$u _ { n + 2 } + \alpha u _ { n + 1 } + \beta u _ { n } = 20 ( - 3 ) ^ { n } \quad n \geqslant 0$$ is given by $$u _ { n } = A ( 2 ) ^ { n } + B ( - 1 ) ^ { n }$$ where \(A\) and \(B\) are arbitrary non-zero constants.

1. Find the value of \(\alpha\) and the value of \(\beta\). Given that \(2 u _ { 0 } = u _ { 1 }\) and \(u _ { 4 } = 164\)
2. find the solution of this second order recurrence relation to obtain an expression for \(u _ { n }\) in terms of \(n\).
   (6)

2. The general solution of the second order recurrence relation $$u _ { n + 2 } + k _ { 1 } u _ { n + 1 } + k _ { 2 } u _ { n } = 0 \quad n \geqslant 0$$ is given by $$u _ { n } = ( A + B n ) ( - 3 ) ^ { n }$$ where \(A\) and \(B\) are arbitrary non-zero constants.

1. Find the value of \(k _ { 1 }\) and the value of \(k _ { 2 }\) Given that \(u _ { 0 } = u _ { 1 } = 1\)
2. find the value of \(A\) and the value of \(B\).