#OK to post homework
#Deven Singh, 09/05/2021, Assignment 1
#Problem 1
#The recursion is R(0)= c0,R(1) = c1,R(2) = c2, R(3) = p1 * R(2) +p2*R(1) +p3*R(0),..., R(n) = p1*R(n - 1) + p2*R(n - 2) + p3*R(n - 3)
#R(4) =  p1*R(3)+p2*R(2)+p3*R(1) = p1*(p1*c2+p2*c1+p3*c0)+p2*c2+p3*c1
#Problem 2
F := proc(p1, p2, p3, c0, c1, c2, n) option remember:
if n = 0 then
  c0:
elif n = 1 then 
  c1: 
elif n = 2 then 
  c2: 
else 
  expand(p1*F(p1, p2, p3, c0, c1, c2, n - 1) + p2*F(p1, p2, p3, c0, c1, c2, n - 2) + p3*F(p1, p2, p3, c0, c1, c2, n - 3))
fi:
end:
F(1, 1, 1, 1, 1, 1, 4);
5
#Problem 3
# (i) extinction
evalf(seq(F(1/4,1/4,1/4,1,1,1,i),i=1..10));
evalf(seq(F(1/4,1/4,1/4,1,1,1,i),i=990..1000));
# (ii) stable population
evalf(seq(F(1/3,1/3,1/3,1,1,1,i),i=1..10));
evalf(seq(F(1/3,1/3,1/3,1,1,1,i),i=990..1000));
# (iii) population explosion
evalf(seq(F(1/2,1/2,1/2,1,1,1,i),i=1..10));
evalf(seq(F(1/2,1/2,1/2,1,1,1,i),i=990..1000));