8.01f First-order recurrence: solve using auxiliary equation and complementary function

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.

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

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.

1. On Jim's 11 th birthday his parents invest \(\pounds 1000\) for him in a savings account.

The account earns 2\% interest each year.
On each subsequent birthday, Jim's parents add another \(\pounds 500\) to this savings account.
Let \(U _ { n }\) be the amount of money that Jim has in his savings account \(n\) years after his 11th birthday, once the interest for the previous year has been paid and the \(\pounds 500\) has been added.

1. Explain, in the context of the problem, why the amount of money that Jim has in his savings account can be modelled by the recurrence relation of the form $$U _ { n } = 1.02 U _ { n - 1 } + 500 \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$
2. State an assumption that must be made for this model to be valid.
3. Solve the recurrence relation $$U _ { n } = 1.02 U _ { n - 1 } + 500 \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$ Jim hopes to be able to buy a car on his 18th birthday.
4. Use the answer to part (c) to find out whether Jim will have enough money in his savings account to buy a car that costs \(\pounds 4500\)

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 .

4. A village has an expected population growth rate (birth rate minus death rate) of \(r \%\) per year. In addition, \(N\) people are expected to move into the village each year. The expected population of the village is modelled by $$u _ { n + 1 } = 1.02 u _ { n } + 50$$ where \(u _ { n }\) is the expected population of the village \(n\) years from now.

1. State
   1. the value of \(r\),
   2. the value of \(N\). Given that the population 1 year from now is expected to be 560
2. solve the recurrence relation for \(u _ { n }\)
3. Hence determine, using algebra, the number of years from now when the model predicts that the population of the village will first be greater than 3000
   (Total for Question 4 is 10 marks)
   TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS END

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 \(u _ { n }\)

7. Martina decides to open a bank account to help her to save for a holiday. Each month she puts \(\pounds \mathrm { k }\) into the account and allows herself to spend one quarter of what was in the account at the end of the previous month. Let \(u _ { n }\) (where \(\mathrm { n } \geqslant 1\) ) represent the amount in the account at the end of month n .
Martina has \(\pounds \mathrm {~K}\) in the account at the end of the first month.

1. By setting up a first order recurrence relation for \(u _ { n + 1 }\) in terms of \(u _ { n }\), determine an expression for \(u _ { n }\) in terms of n and k At the end of the 8th month, Martina needs to have at least \(\pounds 1750\) in the account to pay for her holiday.
2. Determine, to the nearest penny, the minimum amount of money that Martina should put into the account each month.

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.

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]

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]