Recurrence relation solving for closed form

6 A sequence is defined by $$a _ { n + 1 } = 2 a _ { n } + 3 a _ { n - 1 } \quad \text { with } a _ { 1 } = 1 \text { and } a _ { 2 } = 1 .$$ Using the method on page 5, show that the value to which the ratio of successive terms converges is 3 .
[0pt] [4]

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.

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. A recurrence system is defined by

$$\begin{aligned} u _ { n + 2 } & = 9 ( n + 1 ) ^ { 2 } u _ { n } - 3 u _ { n + 1 } \quad n \geqslant 1 \\ u _ { 1 } & = - 3 , u _ { 2 } = 18 \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { N }\), $$u _ { n } = ( - 3 ) ^ { n } n !$$

6. (a) Determine the general solution of the recurrence relation $$u _ { n } = 2 u _ { n - 1 } - u _ { n - 2 } + 2 ^ { n } \quad n \geqslant 2$$ (b) Hence solve this recurrence relation given that \(u _ { 0 } = 2 u _ { 1 }\) and \(u _ { 4 } = 3 u _ { 2 }\)

1. Determine a closed form for the recurrence relation

$$\begin{aligned} & u _ { 0 } = 1 \quad u _ { 1 } = 4 \\ & u _ { n + 2 } = 2 u _ { n + 1 } - \frac { 4 } { 3 } u _ { n } + n \quad n \geqslant 0 \end{aligned}$$

1. Determine a closed form for the recurrence system

$$\begin{gathered} u _ { 1 } = 4 \quad u _ { 2 } = 6 \\ u _ { n + 2 } = 6 u _ { n + 1 } - 9 u _ { n } \quad ( n = 1,2,3 , \ldots ) \end{gathered}$$

1. Sequences \(\left\{ x _ { n } \right\}\) and \(\left\{ y _ { n } \right\}\) for \(n \in \mathbb { N }\), are defined by

$$\begin{gathered} x _ { n + 1 } = 2 y _ { n } + 3 \quad \text { and } \quad y _ { n + 1 } = 3 x _ { n + 1 } - 4 x _ { n } \\ x _ { 1 } = 1 \quad \text { and } \quad y _ { 1 } = a \end{gathered}$$ where \(a\) is a constant.

1. Show that \(x _ { n + 2 } - 6 x _ { n + 1 } + 8 x _ { n } = 3\)
2. Solve the second-order recurrence relation given in (a) to obtain an expression for \(x _ { n }\) in terms of \(a\) and \(n\). Given that \(x _ { 7 } = 28225\)
3. find the value of \(a\).

5. A sequence \(\left\{ u _ { n } \right\}\), where \(\mathrm { n } \geqslant 0\), satisfies the second order recurrence relation $$u _ { n + 2 } = \frac { 1 } { 2 } \left( u _ { n + 1 } + u _ { n } \right) + 3 \text { where } u _ { 0 } = 15 \quad u _ { 1 } = 20$$

1. By considering the sequence \(\left\{ v _ { n } \right\}\), where \(u _ { n } = v _ { n } + 2 n\) for \(\mathrm { n } \geqslant 0\), determine an expression for \(u _ { n }\) as a function of n .
2. Describe the long-term behaviour of \(u _ { n }\)

8. A sequence \(\left\{ u _ { n } \right\}\), where \(n \geqslant 0\), satisfies the recurrence relation $$2 u _ { n + 2 } + 5 u _ { n + 1 } = 3 u _ { n } + 8 n + 2$$

1. Find the general solution of this recurrence relation.
   (5) A particular solution of this recurrence relation has \(u _ { 0 } = 1\) and \(u _ { 1 } = k\), where \(k\) is a positive constant. All terms of the sequence are positive.
2. Determine the value of \(k\).
   (3)

1. (a) Find the general solution of the recurrence relation

$$u _ { n + 2 } = u _ { n + 1 } + u _ { n } , \quad n \geqslant 1$$ Given that \(u _ { 1 } = 1\) and \(u _ { 2 } = 1\) (b) find the particular solution of the recurrence relation.

12

1. Let \(I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 } .$$
2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
   1. Sketch this curve.
   2. Find the exact length of the curve.

12

1. The sequence \(\left\{ u _ { n } \right\}\) is defined for all integers \(n \geq 0\) by $$u _ { 0 } = 1 \quad \text { and } \quad u _ { n } = n u _ { n - 1 } + 1 , \quad n \geq 1 .$$ Prove by induction that \(u _ { n } = n ! \sum _ { r = 0 } ^ { n } \frac { 1 } { r ! }\).
2. (a) Given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x } \mathrm {~d} x\) for \(n \geq 0\), show that, for \(n \geq 1\), $$I _ { n } = n I _ { n - 1 } - \frac { 1 } { \mathrm { e } }$$ (b) Evaluate \(I _ { 0 }\) exactly and deduce the value of \(I _ { 1 }\).
   (c) Show that \(I _ { n } = n ! - \frac { u _ { n } } { \mathrm { e } }\) for all integers \(n \geq 1\).

The sequence \(u_1, u_2, u_3, \ldots\) is defined by \(u_1 = 2\) and \(u_{n+1} = \frac{u_n}{1 + u_n}\) for \(n \geq 1\).

1. Find \(u_2\) and \(u_3\), and show that \(u_4 = \frac{2}{7}\). [3]
2. Hence suggest an expression for \(u_n\). [2]
3. Use induction to prove that your answer to part (ii) is correct. [5]

The sequence \(u_0, u_1, u_2, \ldots\) satisfies the recurrence relation \(u_{n+2} - 3u_{n+1} - 10u_n = 24n - 10\).

1. Determine the general solution of the recurrence relation. [6]
2. Hence determine the particular solution of the recurrence relation for which \(u_0 = 6\) and \(u_1 = 10\). [3]
3. Show, by direct calculation, that your solution in part (b) gives the correct value for \(u_2\). [1]

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). You are given that \(v_n\) converges to a finite non-zero limit, \(q\), as \(n \to \infty\).

1. Determine \(p\) and \(q\). [4]

1. Find the general solution of $$u_n = 8u_{n-1} - 16u_{n-2}, \quad n \geq 2. \quad (*)$$ [4]

A new sequence \(v_n\) is defined by \(v_n = \frac{u_n}{u_{n-1}}\) for \(n \geq 1\).

1. (A) Use (*) to show that \(v_n = 8 - \frac{16}{v_{n-1}}\) for \(n \geq 2\). [2] (B) Deduce that if \(v_n\) tends to a limit then it must be 4. [2]
2. Use your general solution in part (i) to show that \(\lim_{n \to \infty} v_n = 4\). [3]
3. Deduce the value of \(\lim_{n \to \infty} \left(\frac{u_n}{u_{n-2}}\right)\). [1]

A sequence \(t_1, t_2, t_3, t_4, t_5, \ldots\) is given by $$t_{n+1} = at_n + 3n + 2, \quad t \in \mathbb{N}, \quad t_1 = -2,$$ where \(a\) is a non zero constant.

1. Given that \(\sum_{r=1}^{3} (r^3 + t_r) = 12\), determine the possible values of \(a\). [4]
2. Evaluate \(\sum_{r=8}^{31} (t_{r+1} - at_r)\). [4]

The members of the family of the sequences \(\{u_n\}\) satisfy the recurrence relation $$u_{n+1} = 10u_n - u_{n-1} \text{ for } n \geq 1. \quad (*)$$

1. Determine the general solution of (*). [3]
2. The sequences \(\{a_n\}\) and \(\{b_n\}\) are members of this family of sequences, corresponding to the initial terms \(a_0 = 1\), \(a_1 = 5\) and \(b_0 = 0\), \(b_1 = 2\) respectively.
   1. Find the next two terms of each sequence. [1]
   2. Prove that, for all non-negative integers \(n\), \((a_n)^2 - 6(b_n)^2 = 1\). [8]
   3. Determine \(\lim_{n \to \infty} \frac{a_n}{b_n}\). [2]