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Recurrence relations, compound interest, polynomials, number of combinations

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#20
Question The jabby bird is in danger of being placed on the endangered species list, as there are only 975 of the birds known to be in existence. A bird is placed on the list when the known population reaches 100. If 27% of the jabby bird population either dies or is taken by a poacher each year and only 5 new jabby birds are born, write a recurrence relation and initial conditions describing the number jn of jabby birds at the end of n years.

#26
Zebra mussels are fresh water mollusks that attack underwater structures. Suppose that the volume of mussels in a confined area grows at a rate of 0.2% per day.
(a) If there are now 10 cubic feet of mussels in a lock on the Illinois River at Peoria, Illinois, develop a recurrence relation and initial conditions that represent the volume of the mussel colony n days hence.
(b) Develop a formula expressing as a function of n.

#6
Find an explicit formula for if , , ,... is a sequence satisfying the given recurrence relation and initial conditions.

#17
Question: = , = 1, = 9

#28
Suppose that a corporate executive deposits $2000 per year for 35 years into an individual retirement account. If interest is compounded annually at the rate of 8%, how much will the account be worth after the last deposit?

#4
A= 1 + x +
C= 1 - +
Question: AC

#15
In Exercises 13-22, give a generating function for the sequence { }, and then write it in the form + x+ + + ... through the term. Question: Let be the number of ways of choosing r jellybeans from a basket containing 3 licorice, 4 strawberry, and 2 lemon jellybeans.

#18
Question: Let be the number of ways of spending r dollars on posters, if four identical $1 and three identical $2 posters are available.

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#20
Question: The jabby bird is in danger of being placed on the endangered species list, as there are only 975 of the birds known to be in existence. A bird is placed on the list when the known population reaches 100. If 27% of the jabby bird population either dies or is taken by a poacher each year and only 5 new jabby birds are born, write a recurrence relation and initial conditions describing the number jn of jabby birds at the end of n years.

Currently, it is year n = 0 and there are J = 975 jabby birds. Therefore, the initial condition is

J0 = 975

Every year 27% of the population dies. This can be indicated by subtracting 0.27Jn-1 (0.27 times the population of the previous year). The 5 new birds can be indicated by adding 5.

Therefore, the number of jabby birds can be determined using the following recurrence relation:

Jn = Jn-1 - 0.27Jn-1 + 5

This can be simplified as:

Jn = 0.73Jn-1 + 5

This formula means that the population in the year n is calculated by multiplying the previous year's population by 0.73, then adding 5.

#26
Zebra mussels are fresh water mollusks that attack underwater structures. Suppose that the volume of mussels in a confined area grows at a rate of 0.2% per day.

(a) If there are now 10 cubic feet of ...

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