help please

[Guest 3.14lwy]

how to show this:

 

if f(x) = 2x/(2x-1)

 

then show f(n) + f(n)f(n-1) + f(n)f(n-1)f(n-2) + ... + f(n)f(n-1)f(n-2)...f(1) = 2n .

 

 

I have tried to use induction , and I can show it.

but , I can't show it without using induction.

 

 

can anyone tell how to show it without using induction?

 

thank you !

[Dave]

Induction is the most logical step to use to prove something like this - why not use it?

[Guest 3.14lwy]

yes, you are right.

 

but i want to learn more higher skill method ,

 

 

 

just like

 

(sin@ + icos@ )^n = sin(n@) + icos(n@)

 

we can show this by induction, but i think it has a big problem ,

 

if I only learn the induction prove and

 

if I was asked to find (sin@ + icos@ )^n in term of sin(n@) and cos(n@)

 

then i cannot use induction and don't know how to do .

[MandrakeRoot]

Induction is a valid method of proof. Why not use it ?

There is nothing low skill about that method, it is just an elegant way of proving things. If there are other ways you are free to explore them. A transparent proof can be quite handy though and induction usually leads to quite transparents proofs.

 

In the above question you can probably manipulate all terms and do a direct proof though, but the induction proof is probably more insightfull.

 

Mandrake

[Manifold]

I'm not quite familiar with (in)finite sums and products...but I think it has something to do with it...and I guess one might be able to apply them to this problem...

[WaR]

Well, lower skill or higher skill is irrelevant in this case. If you can use induction in a few steps use it. If you want to do the same exercise requiring 10 pages of proof, then you are missing the point of the question.

 

Mathematics wants to make life easier for you, and higher methods are developed for higher more complex problems. Use what you must efficiently and later you will come to problems that require another approach. Just my opinion.

[Guest 3.14lwy]

Sorry to all

 

maybe i had said something wrong before ,

 

i think i should not said this words

"i want to learn more higher skill method "

 

I do not mean MI ( induction ) is a low skill skill method ,

 

I just want to learn more skill .

[MandrakeRoot]

Dont be sorry. Nobody was offended.

 

It is hard to teach general "proof methods", since that doesnt really exists. It really depends on the problem, how you should deroll your proof.

Mostly that will come with time, seeing many proofs and doing many of them.

 

You could try attacking the above problem directly though, bringing all terms into one big fraction and seeing what you can do about it. But personally i dont think that will be the most elegant way to demonstrate the equality.

 

Mandrake