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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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#### Solution Preview

#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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## Recurrence relations, compound interest, polynomials

Please help with the following discrete math involving recurrence relations, compound interest, polynomials, number of combinations and iteration.

14. Individual membership fees at the evergreen tennis club were \$50 in 1970 and have increased by \$2 per year since then. Write a recurrence relation and initial conditions for the membership fee n years after 1970

2. Prove by mathematical induction that 4(2n) +3 is a solution to the recurrence relation sn = 2sn-1 -3 for n ≥ 1 with the initial condition s0 = 7

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