# EXERCISES

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```					                                          EXERCISES

1.       Find the first 5 terms of the sequence defined by the recurrence relation and the
initial condition: a n  4a n 1 , a 0  1
2.       Find a solution to the recurrence relation a n  3a n 1 , a 0  10 . Use an iterative
approach.
3.       A person deposits \$1200 in an account that yields 8.5% interest compounded
yearly.

a)      Set up a recurrence relation for the amount in the account at the end of n years
b)      Find an explicit formula for the amount in the account at the end of n years
c)      How much will the account have after 12 years?

3.        Assume the population of the world in 2000 is 5.9 billion and is growing at the
rate of 1.27% a year.

a)      Set up a recurrence relation for the population n years after 2000.
b)      Find an explicit formula for the population n years after 2000.
c)      What will the population of the world be in 2020?

4.      Solve the recurrence relations:

a)      a n  a n 1  6a n  2 for n  2, a0  3, a1  5
b)      a n  7a n 1  10 a n 2 for n  2, a0  2, a1  2
c)      a n  6a n 1  8a n 2 for n  2, a0  3, a1  10

5.     Consider the discrete logistic recurrence relation x n  rx n 1 (1  x n 1 ), n  1 . Use
Maple to do the following:

a)      Compute x 200 with x1  0.5 and 1  r  3 . Do the results support the conjecture
that x  always exists and is an increasing function of r?
b)      Repeat with r  3.1, r  3.25, r  3.55 . Describe your observations.
c)      Use 2.9  r  3.1 to determine as closely as possible just where the single
limiting population splits into a cycle of period 2.
d)      Find the period of cycle obtained with growth rates of r  3.365 , and r  3.57 .

6.      Suppose we have a population of critters who never grow past five years old.
Suppose, further, that 80% of the critters survive each year to move onto the next age
class, with the exception of the five year old critters which always die. If newborns and
yearlings have no offspring, and if middle aged critters (2, 3 and 4 year olds) have, on
average, 0.35 newborns each year, while the oldest critters (5 year olds) have only 0.1
newborns on average. The initial population consists of sixty critters evenly spread
throughout the six age classes. Determine the population distribution 10 years into the
future.
7.      The following table gives the population of females in six specific age groups of a
small woodland mammal.
Table 1: Female Population of Small Woodland
Mammals
Age          0- 3- 6- 9-
12-15 15-18
(in months)       3 6 9 12
Number of
14 8 12 4         0      0
Females
Table 2: Birth, and Survival Rates for Each Age
Group
Age           0- 3-       9- 12-
6-9            15-18
(in months)       3 6        12 15
Birth Rate        0 0.3 0.8 0.7 0.4         0
Survival Rate     0.6 0.9 0.9 0.8 0.6        0

Find the population after 12 months.

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