Consider the sequence 2, 11, 20, 29, \ldots
Is this an arithmetic or geometric sequence?
State the common difference or ratio.
Consider the sequence 81, - 27 , 9, - 3 , \ldots
Is this an arithmetic or geometric sequence?
State the common difference or ratio.
Is the sequence 3, - 15 , 75, - 300 , \ldots an arithmetic or geometric sequence?
Consider the sequence 2, 7, 17, 37, \ldots
What type of sequence is this?
Write the recursive rule for the sequence, T_{n + 1} in terms of T_{n}, where T_1 = 2.
Consider the sequence 3, -1, 7, - 9 , 23, \ldots
What type of sequence is this?
Write the recursive rule for the sequence, T_{n + 1} in terms of T_{n}, where T_1 = 3.
Write the first five terms of the following recurrence relations:
a_0 = 2 and a_n = 4 a_{n - 1} - 3
a_0 = 2 and a_{n+1} = - 0.5 a_n + 5
For each of the following recurrence relations:
List the first three terms.
Complete a table of values for n=0 to n = 3.
Graph the relation.
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 the first four terms.
Graph the relation.
Determine the largest value of n for which u_n<22.
Consider the sequence 2, 8, 26, 80, \ldots Write a recursive rule for a_n in terms of a_{n - 1} and an initial condition for a_1.
For each difference equation below, determine the value of a_1:
a_n = 5 a_{n - 1} - 18 where a_7 = - 68745.5
a_{n+1}= - 3 a_n + 12 where a_5 = 84
a_n = - 5 a_{n - 1} - 4 where a_6 = - 20834
a_{n+1} = 0.5 a_n - 4 where a_4 = - 7.375
We can define the term values of a sequence by a recurrence relation of the form t_{n + 1} = r t_n + d, where t_1 = a. If - 1 < r < 1, then the term values approaches a steady-state solution in the long term, when n becomes large.
State whether the following recurrence relations define sequences that have a long-term steady-state solution:
t_{n + 1} = - 5 t_n - 5 where t_1 = 6
t_{n + 1} = - 0.6 t_n + 7 where t_1 = - 2
t_{n + 1} = 0.9 t_n - 2 where t_1 = 7
t_{n + 1} = - 2 t_n + 6 where t_1 = - 5
Consider the recurrence relation t_{n + 1} = 0.1 t_n - 9 where t_1 = - 3.
Describe the long term behaviour of the sequence.
State whether each of the following statements about the steady-state is always true:
At the steady-state, t_{n + 1} = t_n.
At the steady-state, the value of the terms reach a stationary point and then begin to decrease.
At the steady-state, n + 1 = n.
At the steady-state, the value of each term is zero.
Determine the steady state solution by setting both t_{n + 1} and t_n to be x.
Consider the recurrence relation t_{n + 1} = - 0.1 t_n - 11 where t_1 = 2.
Describe the long term behaviour of the sequence.
Find the steady state solution by setting both t_{n + 1} and t_n to be x.
Two sequences are defined by the recurrence relations u_{n + 1} = - 0.1 u_n + 55, where u_1 = 2, and v_{n + 1} = 1.3 v_n - 7, where v_1 = 6.
Which recurrence relation, u_{n + 1} or v_{n + 1}, defines a sequence with a long-term steady-state solution?
Find the steady-state solution by setting the (n+1)th term and the nth term of this recurrence relation to x.
For the first order linear difference equations, use your calculator to view the first 100 terms numerically and graphically, and then answer the following questions:
The sequence defined by the recurrence relation u_{n + 1} = k u_n - 16, where u_1 = 3, approaches a long-term steady-state value of - 40. Find the value of k.
A sequence is defined by the recurrence relation t_{n + 1} = k t_n + 7 k, where t_1 = - 2, and k is a real constant. If the long-term steady-state solution is 28, find the value of k. Give your answer as a decimal.
The population of wallabies in a reserve was initially b_0 = 180. Each year the population naturally decreases by 20\%, but then 140 new wallabies are introduced. Let b_n be the population of wallabies after n years.
State the recurrence relation that finds b_{n + 1} in terms of b_n to describe the population of wallabies.
Find the long-term steady-state population of wallabies in the reserve by letting
b_{n + 1} = b_n=x.
What will the population of wallabies in the reserve be in the long-term?
The number of trees in a state forest decreases by 25\% each year due to logging. Before the logging commenced there were 2000 trees in the forest. Each year, in response to the logging, an environment protection group plants 100 new trees in the forest.
Let t_n be the number of trees in the year n.
Write a recurrence relation in terms of t_{n+1} to describe the number of trees in the state forest.
Find the long-term tree population in the state forest by letting t_{n + 1} = t_n = x.
What will the number of trees in the forest be in the long-term?
In a state in India the wild tiger population is dropping by a certain percentage every year. The current population is 250. Each year 10 new tigers are introduced to the tiger reserves in the state. The population of tigers in the state can be modelled by the following first order linear recurrence relation.
T_{n + 1} = 0.935 T_n + b,\text{ } T_1 = 250, where T_n is the number of tigers in the state at the beginning of the nth year
What is the rate of the tiger decline each year as a percentage?
State the value of b.
Find the number of tigers in the state at the beginning of the 10th year. Round your answer to the nearest whole number.
Determine the long term population. Round your answer to the nearest whole number.
In a state in America the black bear population is dropping by a certain percentage every year. The current population is 54. Each year, 12 black bears are introduced to the bear reserves in the state. The population of black bears in the state can be modelled by the following first order linear recurrence relation.
B_{n + 1} = 0.9 B_n + b,\text{ } B_1 = 54, where B_n is the number of black bears in the state at the beginning of the nth year
What is the rate of the black bear decline each year as a percentage?
State the value of b.
Find the number of black bears in the state at the beginning of the 11th year. Round your answer to the nearest whole number.
Determine the long term population. Round your answer to the nearest whole number.
In a province in China the panda population is dropping by a certain percentage every year. The current population is 102. Each year, 4 pandas are introduced to the panda reserves in the province. The population of pandas in the province can be modelled by the following first order linear recurrence relation.
P_{n + 1} = 0.98 P_n + b,\text{ } P_1 = 102, where P_n is the number of pandas in the province at the beginning of the nth year.
What is the rate of the panda decline each year as a percentage?
State the value of b.
Find the number of pandas in the state at the beginning of the 12th year. Round your answer to the nearest whole number.
Determine the long term population. Round your answer to the nearest whole number.
In Alaska the polar bear population is dropping by a certain percentage every year. The current population is 460. Each year, 15 polar bears are introduced to the bear reserves in Alaska. The population of polar bears in Alaska can be modelled by the following first order linear recurrence relation.
P_{n + 1} = 0.97 P_n + b,\text{ } P_1 = 460, where P_n is the number of polar bears in Alaska at the beginning of the nth year.
What is the rate of the polar bear decline each year as a percentage?
State the value of b.
Determine the number of polar bears in Alaska at the beginning of the 9th year. Round your answer to the nearest whole number.
Find the long term population. Round your answer to the nearest whole number.
Emma buys 25 worms to feed her goldfish. She places them in the freezer and feeds her fish 2 worms every week. She also buys 5 new worms at the start of each week. A certain percentage of the worms get freezer burn and are not able to be used.
The number of worms at the start of the nth week can be modelled by W_n, where W_{n + 1} = 0.75 \left(W_n - 2\right) + 5, W_1 = 25
What percentage of worms get freezer burn each week?
Approximately how many worms does Emma have at the start of week 7? Round your answer to the nearest whole number.
Emma claims that she will never run out of worms using this model. Determine the number of worms, x , for a steady state solution to this problem.
Emma is concerned that the constant number of worms won't be enough for her growing fish. Instead, she changes the model to W_{n + 1} = 0.75 \left(W_n - 3\right) + d.
If, over time, she would rather have a constant 23 worms per week, determine the value of d.
Scott buys 35 coffee pods to start his coffee collection. He puts them in a jar and uses 25 pods each week. He also buys 30 new pods at the start of each week. A certain percentage of the pods are faulty and are not able to be used.
The number of pods at the end of the nth week can be modelled by P_n, where P_n = 0.95 \left(P_{n - 1} - 25\right) + 30,\text{ } P_0 = 35
What percentage of coffee pods are faulty each week?
Approximately how many coffee pods does Scott have at the end of week 10? Round your answer to the nearest whole number.
Scott claims that he will never run out of coffee pods using this model. Determine the number of pods, x, for a steady state solution to this problem.
Scott is concerned that the constant number of pods is too high. Instead, he changes the model to P_n = 0.95 \left(P_{n - 1} - 20\right) + d.
If, over time, he would rather have a constant 40 pods per week, find the value of d.
Sarah buys treats to feed her guinea pig. She puts them in the pantry and feeds her guinea pig a certain number every week. She also buys some new treats at the start of each week. A certain percentage of the treats get eaten by pantry mice and are not able to be used.
The number of treats at the start of the nth week can be modelled by T_n, where T_{n + 1} = 0.65 \left(T_n - 6\right) + 10, T_1 = 25
How many treats did Sarah have initially?
How many treats does the guinea pig each eat week?
How many new treats does Sarah buy each week?
What percentage of the treats are eaten by pantry mice each week?
Determine the steady state number of treats T in the pantry. Round your answer to the nearest whole number.
Ann is given 5 cups of Kefir, a yoghurt-like culture that is beneficial for health. She grows the culture by a certain percentage each week and drinks a number of cups per week. Ann also buys some extra cups of Kefir at the end of each week.
The number of cups of Kefir at the of the nth week can be modelled by K_n, where K_{n + 1} = 1.05 \left(K_n - 14\right) + 4, \text{ } K_0 = 5
How many cups of Kefir does Ann drink each week?
By what percentage is the Kefir increasing each week?
How many new cups does Ann buy each week?