Confused About Fibonacci Sequence Problem?

[noboost4you]

My professor really threw a curve ball at me the other day with this problem, and unfortuneately, I have no idea where to begin. Any help would be greatly appreciated.

In the following problem, all the n's are subdomains of the leading coefficient, along with the f#. f(sub 0), f(sub 1), etc..f(sub n+2), f(sub n+1), f(sub n)...a(sub n)


Let fn be the Fibonacci sequence, i.e.:
{f0 = f1 = 1
{fn+2 = fn+1 + fn, for every n >= 0

Define now a new sequence, {an}, given by an = fn+1 / fn

(a) show that an+1 = 1 + (1/an)
(b) assuming that {an}infinity n=1 is convergent, find lim(n->inf) an.

Please offer any kind of assistance you can. Thanks

 

[Muzza]

The first one is just basic algebra. You were given a formula for a_n, use it and see if you can "transform" the LHS into something which looks like the RHS.

 

[matt grime]

If a_n onverges to a, say, then take limits in the equation in part (a) above to find a.

 

[marcus]

My professor really threw a curve ball at me the other day with this problem, and unfortuneately, I have no idea where to begin. Any help would be greatly appreciated.

In the following problem, all the n's are subdomains of the leading coefficient, along with the f#. f(sub 0), f(sub 1), etc..f(sub n+2), f(sub n+1), f(sub n)...a(sub n)


Let fn be the Fibonacci sequence, i.e.:
{f0 = f1 = 1
{fn+2 = fn+1 + fn, for every n >= 0

Define now a new sequence, {an}, given by an = fn+1 / fn

(a) show that an+1 = 1 + (1/an)
(b) assuming that {an}infinity n=1 is convergent, find lim(n->inf) an.

Please offer any kind of assistance you can. Thanks

noboost4you I will give you a piece of advice that might help a lot (or might not depending on you)

if you aint no math whiz then whenever possible (time permitting) experiment with real numbers and a calculator

(dont let them force you to think abstractly and generally before youre ready)

the Fibs are:
1,1,2,3,5,8,13,21,...


he wants you to study the ratios

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13,...

he wants you to check that

this sequence approaches a number X with the special
nice feature that
X = 1 + 1/X

multiplying thru by X you see that another way to write that is
X² = X + 1

it wouldn't be true exactly for 21/13, or for anyone ratio, but
it might be almost true for 21/13

and if you crank out some larger Fibs and take a corresponding ratio furtherout in the sequence it should be closer to being true for that

If you can figure what number satisfies the equation exactly
that is what X has
X² = X + 1
exactly
then you can tell what the successive ratios of Fibs are going to get closer and closer to

he doesn't want this for an answer, he has a special fancypants way of finding it out
which you are supposed to step thru, like a trained poodle in the circus.
but this is the gist of it
the successive ratios in the Fib sequence go to the Golden Mean
and you can find this out with a ten buck calculator

 

[noboost4you]

The first one is just basic algebra. You were given a formula for a_n, use it and see if you can "transform" the LHS into something which looks like the RHS.

still lost, I am sorry. any other pointers?

 

[noboost4you]

noboost4you I will give you a piece of advice that might help a lot (or might not depending on you)

if you aint no math whiz then whenever possible (time permitting) experiment with real numbers and a calculator

(dont let them force you to think abstractly and generally before youre ready)

the Fibs are:
1,1,2,3,5,8,13,21,...


he wants you to study the ratios

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13,...

he wants you to check that

this sequence approaches a number X with the special
nice feature that
X = 1 + 1/X

multiplying thru by X you see that another way to write that is
X² = X + 1

it wouldn't be true exactly for 21/13, or for anyone ratio, but
it might be almost true for 21/13

and if you crank out some larger Fibs and take a corresponding ratio furtherout in the sequence it should be closer to being true for that

If you can figure what number satisfies the equation exactly
that is what X has
X² = X + 1
exactly
then you can tell what the successive ratios of Fibs are going to get closer and closer to

he doesn't want this for an answer, he has a special fancypants way of finding it out
which you are supposed to step thru, like a trained poodle in the circus.
but this is the gist of it
the successive ratios in the Fib sequence go to the Golden Mean
and you can find this out with a ten buck calculator

sorry, i posted my last message before i saw your response. and to tell you the truth, your response helped me a lot. i understand what my professor is now asking. thanks

 

[marcus]

sorry, i posted my last message before i saw your response. and to tell you the truth, your response helped me a lot. i understand what my professor is now asking. thanks

heh heh consider that you just got a boost
now comes the part of figuring out what el proffo
wants to see on dah homework paper
good luck
BTW they are pretty good at helping if you go down
to college level homework help, near the bottom of
the index page, or so I thought when I looked in there
someone named Doc Al, I think
you can keep asking questions and eventually something may click