Recurrence relations with rabbits pairs

[jonroberts74]

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

1) Each pair is not fertile for their first month bet thereafter give birth to four new male/female pairs at the end of every month

2) no rabbits die

a) let $$r_{n}$$ be the number of pairs of rabbits alive at the end of each month n for each integer $$n \ge 1$$ find a recurrence relation for $$r_{0},r_{1},r_{2}...$$

b) how many rabbits will there be at the end of the yearMonth | Babies (in pairs) | Adults (in pairs) | total Pairs (r_{n})
1 |1 |0 |1
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2 |4 |1 |5
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3 |20 |5 |25
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4 |100 |25 |125
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5 |400 |125 |525
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6 |2100 |525 |2625
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7 |10500 |2625 |13125
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8 |52500 |13125 |65625
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9 |262500 |65625 |328125
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10 |1312500 |328125 |1640625
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11 |6562500 |1640625 |8203125
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12 |328125000 |8203125 |41015625

(sorry for the ugly table)

the recurrence relation seems to be $$r_{n} = a+4a$$ where a = number of adults, for $$n \ge 1$$

is that correct?

and part b) would be 41,015,625 pairs so 82,031,250 rabbits

 

[micromass]

the recurrence relation seems to be $$r_{n} = a+4a$$ where a = number of adults, for $$n \ge 1$$

is that correct?

It is not correct. You have now counted the adults and the newborns, but not the rabbits which are now 1 month old.



It is also not in the form they are looking for. You have to express $r_n$ in terms of $r_k$ with $k<n$. So for example, a possible (but obviously incorrect) answer would be $r_n = 100r_{n-1} + 5\sqrt{r_{n-3}r_{n-2}}$.

You express it in terms of $a$. Can you perhaps express $a$ in terms of $r_k$?

 

[jonroberts74]

It is not correct. You have now counted the adults and the newborns, but not the rabbits which are now 1 month old.
It is also not in the form they are looking for. You have to express $r_n$ in terms of $r_k$ with $k<n$. So for example, a possible (but obviously incorrect) answer would be $r_n = 100r_{n-1} + 5\sqrt{r_{n-3}r_{n-2}}$.

You express it in terms of $a$. Can you perhaps express $a$ in terms of $r_k$?

aren't the rabbits that are 1 month old adults now because it says they are not fertile for the first month then give birth to 4 (male/female) pairs at the end of the month.

or am I incorrect in my computation

should it be

in the beginning theres

1 pair born, then the first month they have no births then the second month they make 4 pairs.

then the 3rd month: the original pair breeds 4 , the 4 pairs from the last month grow to maturity for a total of 9 pairs

then in month 4 there's 5 pairs that can breed, 4 pairs that mature, and 20 pairs born

 

[jonroberts74]

adult pairs = adults of previous month + maturing pairs of previous month

maturing pairs = baby pairs from previous month

baby pairs = 4(adult pairs from previous month + maturing pairs from previous month)

 

[jonroberts74]

| month | Adults | babies | maturing | total |
| ----- | ------ | ------ | --------- | ----- |
| 1 | 0 | 0 | 1 | 1 |
| 2 | 1 | 4 | 0 | 5 |
| 3 | 1 | 4 | 4 | 9 |
| 4 | 5 | 20 | 4 | 29 |
| 5 | 9 | 36 | 20 | 65 |
| 6 | 29 | 116 | 36 | 181 |
| 7 | 65 | 260 | 116 | 441 |
| 8 | 181 | 724 | 260 | 1165 |
| 9 | 441 | 1764 | 724 | 2929 |
| 10 | 1165 | 4660 | 1765 | 7589 |
| 11 | 2929 | 11716 | 4660 | 19305 |
| 12 | 7589 | 30356 | 11716 | 49661 |

 

[Nathanael]

| month | Adults | babies | maturing | total |
| ----- | ------ | ------ | --------- | ----- |
| 1 | 0 | 0 | 1 | 1 |
| 2 | 1 | 4 | 0 | 5 |
| 3 | 1 | 4 | 4 | 9 |
| 4 | 5 | 20 | 4 | 29 |
| 5 | 9 | 36 | 20 | 65 |
| 6 | 29 | 116 | 36 | 181 |
| 7 | 65 | 260 | 116 | 441 |
| 8 | 181 | 724 | 260 | 1165 |
| 9 | 441 | 1764 | 724 | 2929 |
| 10 | 1165 | 4660 | 1765 | 7589 |
| 11 | 2929 | 11716 | 4660 | 19305 |
| 12 | 7589 | 30356 | 11716 | 49661 |

Your table seems correct to me (I haven't checked it to the end)

the recurrence relation seems to be $$r_{n} = a+4a$$ where a = number of adults

Try to avoid using "a" (the number of adults), it's unnecessary.

How could you write it in terms of $r_n$? What do all the adult rabbits have in common?

 

[jonroberts74]

I think the recurrence relation is

$$r_{n}=r_{n-2}+4r_{n-2} + r_{n-1} - r_{n-2}$$

which simplifies to

$$4r_{n-2} + r_{n-1}$$

 

[Nathanael]

which simplifies to

$$4r_{n-2} + r_{n-1}$$

Looks good to me!

 

[micromass]

I think the recurrence relation is

$$r_{n}=r_{n-2}+4r_{n-2} + r_{n-1} - r_{n-2}$$

which simplifies to

$$4r_{n-2} + r_{n-1}$$

Seems ok!