(a) Fibonacci posed the following problem: Suppose thatrabbits live forever and that every month each pair producesrabbits live forever and that every

postillan4

postillan4

Answered question

2021-02-25

(a) Fibonacci posed the following problem: Suppose thatrabbits live forever and that every month each pair producesrabbits live forever and that every month each pair produces a newpair which becomes productive at age 2 months. If we start with onenewborn pair, how many pairs of rabbits will we have in the nthmonth? Show that the answer is fn were {fn} is the fibonaccisequence defined in Example 3(c)
(b) Let an=fn+1fn and show that an1=1+1an2 Assuming that an is convergent, find itslimit.

Answer & Explanation

Pohanginah

Pohanginah

Skilled2021-02-26Added 96 answers

a) Let an be the number of rabbit pairs in the nth month. Clearly a+1=1=a+2
In the nth month, each pair that is 2 or more months old (that is a n-2 pairs) will produce a new pair to add to thea n-1 pairs already present.
Thus,an=an1+an2 so that an=fn the Fibonacci sequence. b) an=fn+1fnfnfn1=fn1+fn2fn1=1+fn2fn1
=1+1bfn1fn2=1+1an2
If L=limnan,then L=limnan1
L=limnan2
So L must satisfy
L=1+1LL2L1=0, L=1+52
(since L must be positive)

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