Page 303 - A Course in Linear Algebra with Applications

8.2: Systems of Linear Recurrences 287

2. In a certain nature reserve there are two competing animal
species A and B. It is observed that the number of species A
equals three times the number of A last year less twice the
number of species B last year. Also the number of species B
is twice the number of B last year less the number of species A
last year. Write down a system of linear recurrence relations
for a n and b n, the numbers of each species after n years, and
solve the system. What are the long term prospects for each
species?
3. A pair of newborn rabbits begins to breed at age one
month, and each successive month produces one pair of off-
spring (one of each sex). Initially there were two pairs of rab-
bits. If r n is the total number of pairs of rabbits at the begin-
ning of the nth month, show that r n satisfies r n+i = r n + r n _ i
and ri = 2 = r^- Solve this second order recurrence relation
for r n.
4. A tower n feet high is to be built from red, white and blue
blocks. Each red block is 1 foot high, while the white and
blue blocks are 2 feet high. If u n denotes the number of dif-
ferent designs for the tower, show that the recurrence relation
u n+i = u n + 2u n _i must hold. By solving this recurrence,
find a formula for u n.
5. Solve the system of recurrence relations y n +i = 3y n — 2z n,
z n+i — 2y n — z n, with the initial conditions yo = 1, zo = 0.
6. Solve the second order system y n +i = y n-i, z n+i = y n +
4z n, with the initial conditions yo = 0, y\ = 1 = z\.
7. In a certain city 90% of employed persons retain their jobs
at the end of each year, while 60% of the unemployed find
a job during the year. Assuming that the total employable
population remains constant, find the unemployment rate in
the long run.