1. [6] Recursive formulas. See notes, Friday Oct. 27. Motivate your answers, and
ID: 3147108 • Letter: 1
Question
1. [6] Recursive formulas. See notes, Friday Oct. 27. Motivate your answers, and don’t forget the initial values!.
(b) Fibonacci found a different breed of rabbit, where females are first able to produce offspring in their third month of life, instead of in the second month. Find a recursive formula for the number of female rabbits after n months. Hint: use Fibonacci words and replacement rules as before, but now have three kind of symbols: M for mature rabbits, N for newborn rabbits, I for immature rabbits in their second month.
Explanation / Answer
We have given that, Fibonacci found a different breed of rabbit, where females are first able to produce offspring in their third month of life, instead of in the second month.Our AIM is to find a recursive formula for the number of female rabbits after n months.
The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
(fromthis at the first month and second month the pair of rabbits are 1and1, in the third month this pair becomes(1+1) =2 pairs in the fourth month it becomes (1+2) =3 pairs ),in the fifth month this pair becomes (3+2)=5 pairs and so on after n months the number of pairs .Let N,I1,I2,M1,M2 be the number of pairs of newborn, pairs of one-month old, and pairs of two-month old rabbits respectively after ii months. Pairs in category M1 and M2 produce new pairs (counted in M1+1). After being counted in M2, a pair leaves the island and is not counted in the next iteration.
Let vi=NI1M2vi=(NI1M2)
Initial conditions are
v0=100v0=(100).
For n1n1, we have
Nn=I1n-1+M2n-1
M1n=Nn-1
M2n=M1n-1
Note that this gives v1=010v1=(010) and v2=101v2=(101).
Then the total number of rabbits after n months, for n3 is given by:
an=Nn+I1n+M2n
an=(I1n1+M2n-1)+Nn-1+I1n-1
an=(Nn-2+I1-2)+In-2+M2n-2+Nn-2
an=(Nn-2+I1-2)+Nn-3+M2n-2+I1n-3+M2n-3.
n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ... Fib(n): 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ..More..Related Questions
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