Question

Question #60254

Answer 1

`a_n = 4(a_(n-1))+5`

As far as finding which cycle produced 5000 rabbits, the first cycle where the number of rabbits breaks 5000 is cycle 5 (cycle 4 had 2985 rabbits, and cycle 5 had 11945 rabbits).

Explanation:

So at the beginning of cycle 1 (cycle 0, I guess you could call it), there are 10 rabbits; half of these are females if I've read the question correctly.

Cycle 0: 5 males, 5 females
Total rabbits: 10

During cycle 1 (days 1-50), each of the 5 females produce 4 males and 3 females. This means that the rabbit population gains 20 males and 15 females.

Cycle 1: 25 males, 20 females
Total rabbits: 45

During cycle 2 (days 51-100), each of the 20 females produce 4 males and 3 females. This means that the rabbit population gains 80 males and 60 females.

Cycle 2: 105 males, 80 females
Total rabbits: 185 (at this point, we can look at the numbers and guess what the pattern is, but we may need some more info).

During cycle 3 (days 101-150), each of the 80 females produce 4 males and 3 females. This means that the rabbit population gains 320 males and 240 females.

Cycle 3: 425 males, 320 females
Total rabbits: 745

The pattern is more obvious now; each time, the number of rabbits increases by a factor of 4 and then gains 5 more. In other words, if cycle `n` has `x` rabbits, cycle `n+1` will have `4x+5` rabbits. How do we represent this?

The best way to represent this is to say:
`a_n = 4(a_(n-1))+5`

Also, there isn't a good way to write this as a function, but, as the cycle number grows, this function grows closer and closer to `y = 10(4^x)`.