State whether the following difference equations generate arithmetic sequences:
u_{n} = u_{n - 1} - 9
t_{n + 1} = t_{n} + 5
t_{n + 1} = 8 t_{n}
G_{n + 1} = 6 - G_{n}
t_{n + 1} = 2 t_{n} + 7
For each of the following recurrence relations:
Construct a table of the form:
| n | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| V_{n} |
Graph the relation.
V_{n + 1} = V_{n} + 3 and V_{0} = 10
V_{n + 1} = V_{n} + 100 and V_{0} = 150
V_{n + 1} = V_{n} -8 and V_{0} = 20
Consider the recurrence relation u_{n + 1} = u_{n} + 3.
Complete the table by writing the terms u_{2} to u_{4} in terms of u_{0}:
Express u_7 in terms of u_{0}.
Express u_{n} in terms of u_{0}.
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| u_{n} | u_{0} + 3 |
A piece of artwork purchased for \$2000 increases in value by \$50 per year. Write a recursive rule for V_{n + 1}, the value of the artwork after the \left(n + 1\right) \text{th} year, in terms of V_{n}, and an initial condition for V_{0}.
For a photo, the animals at an animal shelter have been arranged such that there are 10 animals in the front row and each row has 7 more animals than in the row in front of it.
Write a recursive rule for a_{n + 1}, the number of animals in the \left(n + 1\right) \text{th} row, in terms of a_{n}, and an initial condition for a_{0}.
Write a formula for a_{n} in terms of n.
Find the number of animals in the 7th row.
The balance of a savings account earning simple interest each year is given by the explicit rule:V_n = 2100 + 400 \left(n - 1\right)Where V_n (in dollars) is the balance after n years.
How much interest is the account earning each year?
How much is in the account after one year?
What was the original investment amount?
A mobile phone depreciates in value by a constant amount per month and its value is given by the explicit rule:V_n = 1200 - 25 nWhere V_n (in dollars) is the balance after n months.
By how much does the value of the phone depreciate each month?
What was the purchase price of the phone?
State whether the following difference equations generate geometric sequences:
u_{n} = - 2 u_{n - 1}
t_{n + 1} = 0.57 t_{n}
G_{n + 1} = 5 - G_{n}
t_{n + 1} = t_{n} + 9
t_{n + 1} = 6 t_{n}
t_{n + 1} = 8 t_{n} + 7
For each of the following recurrence relations:
Construct and complete a table of the form:
| n | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| V_{n} |
Graph the relation.
V_{n + 1} = 2V_{n} and V_{0} = 3
V_{n + 1} = 0.5V_{n} and V_{0} = 160
V_{n + 1} = 2.5V_{n} and V_{0} = 16
Consider the recurrence relation u_{n + 1}=2 u_{n}.
Complete the table by writing terms u_{2} to u_{4} in terms of u_{0}:
Express u_{7} in terms of u_{0}.
Express u_{n} in terms of u_{0}.
| n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| u_{n} | 2u_{0} |
The average annual rate of inflation in Kazakhstan is 2.6\%. Bread cost \$3.65 in 2015.
What would the bread cost in 2016?
At this rate, what would bread cost in 2018?
Write a recursive rule for V_{n+1}, the cost of bread n+1 years after 2015, in terms of V_{n} and an initial condition V_{0}.
The population of some native bees is declining at a rate of 10\% per year. If there are 23\,700 bees in a hive now, how many will there be in 6 years time?
A bouncy ball is dropped onto the ground from a height of 13 \text{ m}. On each bounce, the ball reaches a maximum height of 50\% of its previous maximum height.
Write a recursive rule for a_{n + 1}, the height of the ball on the \left(n + 1\right) \text{th} bounce, in terms of a_{n}, and an initial condition a_{0}.
Write a formula for a_{n}, for the height reached on the n \text{th} bounce in terms of n.
Find the height that the bouncy ball reaches after the 4th bounce. Round your answer to two decimal places.
Consider the sequence a_{n + 1} = 2 a_{n} + 9. If a_{3} = 31, find a_{1}.
For each of the following recurrence relations:
Complete a table of the form:
Graph the relation.
| n | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| U_{n} |
U_{n + 1} = 0.5 U_{n} + 2 and U_{0} = 20
U_{n + 1} = 2 U_{n} + 100 and U_{0} = - 50
U_{n + 1} = - 2 U_{n} - 4 and U_{0} = 2
Consider the recurrence relation u_{n + 1} = 2 u_{n} + 2, u_{0} = 2.
Find:
Construct a table of values showing u_n for n=0,1,2,3,4.
Graph the relation.
Find the largest value of n for which u_{n} \lt 22 .
Wildlife authorities are trying to control the spread of cane toads in a reserve. Each year there is a 22\% increase in the population size of the cane toads due to breeding and the wildlife authorities aim to capture and remove 500 cane toads each year. At the start of 2015 the cane toad population in the reserve was 2200.
Write a recurrence relation for P_{n+1}, the size of the canetoad population after \left(n+1\right) years, in terms of P_n, and an initial condition P_0.
How many cane toads will be in the reserve by the end of 2018?
Will the removal of 500 cane toads per year ever completely eradicate the cane toad population from this reserve?
Uther’s garden has 5000 weeds in it whose population is increasing at a rate of 0.4\% per month. At the end of each month Uther kills 750 weeds with herbicide.
How many weeds in Uther’s garden at the end of the first month?
Write the recursive rule and initial condition for this situation.
After how many months will Uther have a weed free garden?
The volume of water in a pond in being monitored to ensure that it does not dry out. The pond initially contained 70\,000L but it appears that 20\%of water is lost through evaporation each day. This is compensated for by 2000 L being pumped into the pond each day from a nearby dam.
Write a recurrence relation for V_{n+1}, the volume of water in the pond after \left(n+1\right) days, in terms of V_n and an initial condition V_0.
Determine how many litres of water will be contained in the pond after 20 days. Round your answer to the nearest litre.
How many litres of water will be contained in the pond in the long term?