please help. I\'m lost on what to do. Teacher says births don\'t occur until the

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please help. I'm lost on what to do. Teacher says births don't occur until the fourth month. both questions go together as one

6. [10 points) A single pair of rabbits (male and female) is born at the beginning of a year. Assume the following conditions: (a) Rabbit pairs are not fertile during their first two months of life, but thereafter they give birth to four new male/female pairs at the end of every month; (b) No rabbits ever die. Let sn the number of pairs of rabbits alive at the end of month n, for each integer n 2 1; and let so 1: Find a recurrence relation for so, s1, 2.. Justify your answer carefully. 7. (5 points] In the previous question, how many rabbits are there in month 20? (I recom- mend you model the problem with a short computer program or similar.)

Explanation / Answer

Solution:

The crucial observation is that the number of rabbit pairs born at the end of month k is the same as the number of pairs alive at the end of month k-2. Why? Because it is exactly the rabbit pairs that were alive at the end of month k-2 that were fertile during month k. The rabbits born at the end of month k-1 were not.
so at month k-2, each pair alive
at month k-1 nothing
at month k-2 gives birth to a pair here
F_0 = the initial number of rabbit pairs = 1
and F_1 = 1 also because the first pair of rabbits is not fertile until the 2nd month. Hence the complete specification of the Fibbonacci sequence for all integers k >= 2,
(1) F_k = F_k-1 + F_k-2
(2) F_0 = 1, F_1 = 1

Okay now back to my problem,

The number of rabbits alive at the end of the first month is still 1, so S_0 = 1;
Now it won't be another 2 months after the first month until they are fertile, so at the end of month 1 they still will only have 1 pair, so S_1 = 1, and also at month 2, they still haven't mated, so S_2 = 1, but now they can mate at month 2, so at month 3 (S_3) this is where the babies start popping out. So if its 4(pairs) new male/females, I would just take
4*S_1 right?

S_3 = S_0 + 4*S_1
S_4 = S_1 + 4*S_2
S_5 = S_2 + 4*S_3
...
...
S_k = S_k-3 + 4*S_k-2

7)

similarly, it will continue until 20

Now this recurrence can be implemented in a program.

I hope this helps if you find any problem. Please comment below. Don't forget to give a thumbs up if you liked it. :)

Total number of pairs Fertile Non fertile s0 1 0 1 s1 1 0 1 s2 1 1 0 s3 5 1 4 s4 9 1 10 s5 25 5 20

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