- #1

mr_coffee

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The question is the following:

A single pair of rabbits (male and female) is born at the beginning of a year. Assume the folowing conditions:

1. Rabbit pairs are not fertile during their first

**2 months of life**, but therearfter give birth to

**3 new male/female pairs**at the end of every month.

2. No rabbits die.

a. Let s_n = the number of pairs of rabbits alive at the end of month n, for each integer n>=1, and let s_0 = 1. Find a reuccrence relation for s_0, s_1, s_2...

c. How many rabbits will there be at the end of the year?

answer: 904 rabbit pairs or 1,808 rabits after 12 months.

Well I looked at the orginal Fibonacci's question for a basis:

The orginal question was under the following conditions:

1. Rabbit pairs are not fertile during their

**1st month**of life, but therearfter give birth to

**1 new male/female pairs**at the end of every month.

2. No rabbits die.

How many rabits will there be at the end of the year?

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 intial number of rabbit paris = 1

and F_1 = 1 also, becuase the first apir 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 havn'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 3 new male/females, I would just take

3*S_1 right?

S_3 = S_0 + 3*S_1

S_4 = S_1 + 3*S_2

S_5 = S_2 + 3*S_3

...

...

S_k = S_k-3 + 3*S_k-2

Does that look right for the recurance relationship?

Thanks

Thanks!