Tutor Marked Assignment 5

Recurrence Relations
There are 4 questions in this section. Complete each question and then pass your answers to your tutor for marking.

1. A painting appreciates in value by 15% each year. It is initially worth £1 million.
 
(a) Write down a formula which will allow you to home in on its value after any number of years. Then find its value after (i) 3 years (ii) 5 years.
(b) Write down a recurrence relation which ties the value each year to the value the previous year. How many years is it before the painting trebles in value?

2. At the beginning of November a person borrows £4,000 and agrees to pay back £500 at the beginning of each month after this (i.e. first repayment at the beginning of December). However, during every month the amount owed increases by 2% of the value at the start of the month.
 
(a) Writing down the amount owing every month as you go, calculate during which month the loan will be eventually paid off. In other words, number crunch your way through the problem.
(b) Write down the problem in the form of a recurrence relation which allows you the use of the ANS key.

3. There are an estimated 20 million brownling fish in the sea. During the breeding season there is an estimated 8% increase in the number of fish. After this comes the fishing season and fishermen are allowed to catch 1,750,000 fish.
 
(a) Write down the number of fish in the sea each year in the form of a recurrence relation.
(b) How many fish will be in the sea after four years?
(c) Does the relation have a limit? If so, find it and say what it means; if not, why not? And what are the implications for the fish?

4. In November a turkey farmer has 6,000 turkeys. Each following Christmas he sells off 20% of them and buys in 2,000 chicks which he then brings on for the next season.
 
(a) Write down a recurrence relation for the number of turkeys.
(b) Find the farmer’s long-term position if he continues this policy.