3.07 First order linear recurrence - calculator assumed

Consider the sequence 2, 11, 20, 29, \ldots

Consider the sequence 81, - 27 , 9, - 3 , \ldots

Is the sequence 3, - 15 , 75, - 300 , \ldots an arithmetic or geometric sequence?

Consider the sequence 2, 7, 17, 37, \ldots

Consider the sequence 3, -1, 7, - 9 , 23, \ldots

Write the first five terms of the following recurrence relations:

For each of the following recurrence relations:

Consider the recurrence relation u_{n + 1} = 2 u_n + 2, u_0 = 2.

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:

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:

Consider the recurrence relation t_{n + 1} = 0.1 t_n - 9 where t_1 = - 3.

Consider the recurrence relation t_{n + 1} = - 0.1 t_n - 11 where t_1 = 2.

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.

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.

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.

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

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

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.

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.

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

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

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

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