(a) Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one nowoorn pan v many pairs of rabbits will we have in the nth month? Show that the answer is f, where (f.)}} is the Fibonacci sequence defined recursively by the following conditions. n = fn -1+n- 2 n2 3 From Let r, be the number of rabbit pairs in the nth month. Since we're starting with one newborn pair, r, = 1, and since each pair of rabbits only produces a new pair when the pair is 2 mor hs old, r, - the third month forward, thke original pair of ra bits will begin to produce a new of rabbits each month. f2 = 1 mont will be too young to reproduce, but all rabbits present in the (nth - 2) month can. The number of producingirs each month willed In general, in the nth month, the rabbits born n the (nth - to the pairs already present the previous month Which of the following comple s the proof? O Thus, r.- 1 =n+n-2 Thus, r, = rn -1+n -2 Thus, r, = ro -n - 2 Thus, r,-n- 1-n- 2 %3D Thus, r,-ro+n-2 So {r,) = (). 1 -? 'n +1 Which of the following shows that a,1 =1+ fo (b) Let a,= an-2 %3D 1 fn +1 - an -1 an = -1+n-2 -1+ n-2 -1+ In-1 = 1 + fn-1/fn-2 » an-1 %3D fn -1 fn - 1 1 a, - 2 fo =1+ fn+1 - an -1 an= fo fn+ fn - 1 - 1 + fn - 1 = 1+ an - 2 » an %3D 1-1 fn- 1 1 fn-1+ fn-2 -1+ n-1-1+ - 1+ fn- 2fn -1 = 1+ fn+1 - an -1 -2 an= n-2 an - 2 1 fn - 1 = 1 + n-1 = 1+ fn- 2 = 1 + fn-1- fn+1 - an -1 an fo fn- 2fn-1 + an - fn-1+n-2 1 an - 2 fn+1 - an -1- O a.= fn+2 - n+1+fn - 1+'n = 1 + fn+1 = 1 + ffn+1 » an -1 fn + 1 an- 2 f+1 Assuming that (a,) is onvergent, find its limit. (If an answer does not exist, enter DNE.)

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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(a) Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one nowoorn pa y many
pairs of rabbits will we have in the nth month? Show that the answer is f, where (f.)} is the Fibonacci sequence defined recursively by the following conditions.
fn = fn-1+ fn - 2
n2 3
From
Let r, be the number of rabbit pairs in the nth month. Since we're starting with one newborn pair, r, = 1, and since each pair of rabbits only produces a new pair when the pair is 2 mor hs old, r, -
the third month forward, thke original pair of ra bits will begin to produce a news of rabbits each month.
1 = 1
f2 = 1
mont will be to0 young to reproduce, but all rabbits present in the (nth - 2) month can. The number of producingirs each month wilILed
In general, in the nth month, the rabbits born n the (nth -
to the pairs already present the previous month Which of the following comple s the proof?
O Thus, r.-1 = n+n - 2
Thus, r, =rn -1+n -2
Thus, r= ro-n - 2
Thus, r,-n- 1-n- 2
Thus, r, = ro+n-2
So {r,) = {}.
1
-?
'n + 1.
Which of the following shows that a,-1=1+
fo
(b) Let a, =
an-2
1
fn-1+n-2 = 1 +
fn-2 -1+
fn -1
= 1 +
fn +1 - an -1=
fo
f-1/fn-2
> an-1
%3D
an =
fn - 1
n- 1
1
an - 2
f, n+ fn -1 - 1 +
fn = 1+
= 1+
an - 2
fn+1 - an -1
an=
fn
%3D
fn - 1
fn - 1
1-1
1
fn-1+ fn-2 - 1+ n-1-1+
- 1+
fn-2fn -1
fo
fn+1 - an -1 a- 2
an =
n - 2
an - 2
fn-1-1+
= 1 +
fn- 2
1
= 1 +
fn-1=
fn-1
fn+1 -
an
fo
fn - 2fn -1
+ an - 1
fn-1+n-2
1
an - 2
1
fn+2 - n+1+fn = 1+'n
f+1
1
= 1 +
fn+1 - an -1-
O a.=
= 1 +
fn+1
ffn + 1
» an - 1
fn + 1
an - 2
u.
Assuming that (a,) is onvergent, find its limit. (If an answer does not exist, enter DNE.)
Transcribed Image Text:(a) Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one nowoorn pa y many pairs of rabbits will we have in the nth month? Show that the answer is f, where (f.)} is the Fibonacci sequence defined recursively by the following conditions. fn = fn-1+ fn - 2 n2 3 From Let r, be the number of rabbit pairs in the nth month. Since we're starting with one newborn pair, r, = 1, and since each pair of rabbits only produces a new pair when the pair is 2 mor hs old, r, - the third month forward, thke original pair of ra bits will begin to produce a news of rabbits each month. 1 = 1 f2 = 1 mont will be to0 young to reproduce, but all rabbits present in the (nth - 2) month can. The number of producingirs each month wilILed In general, in the nth month, the rabbits born n the (nth - to the pairs already present the previous month Which of the following comple s the proof? O Thus, r.-1 = n+n - 2 Thus, r, =rn -1+n -2 Thus, r= ro-n - 2 Thus, r,-n- 1-n- 2 Thus, r, = ro+n-2 So {r,) = {}. 1 -? 'n + 1. Which of the following shows that a,-1=1+ fo (b) Let a, = an-2 1 fn-1+n-2 = 1 + fn-2 -1+ fn -1 = 1 + fn +1 - an -1= fo f-1/fn-2 > an-1 %3D an = fn - 1 n- 1 1 an - 2 f, n+ fn -1 - 1 + fn = 1+ = 1+ an - 2 fn+1 - an -1 an= fn %3D fn - 1 fn - 1 1-1 1 fn-1+ fn-2 - 1+ n-1-1+ - 1+ fn-2fn -1 fo fn+1 - an -1 a- 2 an = n - 2 an - 2 fn-1-1+ = 1 + fn- 2 1 = 1 + fn-1= fn-1 fn+1 - an fo fn - 2fn -1 + an - 1 fn-1+n-2 1 an - 2 1 fn+2 - n+1+fn = 1+'n f+1 1 = 1 + fn+1 - an -1- O a.= = 1 + fn+1 ffn + 1 » an - 1 fn + 1 an - 2 u. Assuming that (a,) is onvergent, find its limit. (If an answer does not exist, enter DNE.)
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