(a) Find a recurrence relation for the number of pairs of rabbits on the island
ID: 3825203 • Letter: #
Question
(a)Find a recurrence relation for the number of pairs of rabbits on the island n months after one newborn pair is left on the island. (b) What are the initial conditions for the recurrence relation in part (a)?
6 points (Rosen 8.2.45) Suppose that each pair of a genetically engineered species of rabbits left on an island produces two new pairs of rabbits at the age of 1 month and six new pairs of rabbits at the age of 2 months and every month afterward. None of the rabbits ever die or leave the island. (a) Find a recurrence relation for the number of pairs of rabbits on the island n months after one newborn pair is left on the island. Justify your answer. (b) at are the initial conditions for the recurrence relation in part (a)?Explanation / Answer
According to the question: the numbers will increase each month like below:
Month 1 = 1 + 1*2 = 3
Month 2 = ((1*2)*2 = 4) +( 1*6=6) + 3 = 13
Month 3 = (4+6)*2 + 3*6 + 13 =51
So, In General case the Recurrence Relation would be
T(n)=3T(n-1) + 4T(n-2)
Answer b)
Intial condtion are:
T(-1)=0 and T(0)=1
lets check
for n=1
T(n)=3T(n-1) + 4T(n-2)
T(n)=3T(0)+4(-1)
=>3+0
=>3
for n=2
T(n)=3T(n-1) + 4T(n-2)
=> 3*T(1)+4T(0)
=> 3*3 + 4*1
=>9+4
=>13
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