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Maths - STEP Question Help

Hello,
I’m currently having trouble in attempting this question:
095E6EA9-19F4-49A5-8905-234D57E3EA6C.jpg.jpeg

I’m completely confused as to where to begin in proving part (i) I have tried to take a general graph of y = 1 + (1/x) and sketching it to help me but its of no use as far as I can see, I know from the question that x1, x2, x3, ... , xn cannot equal 0 but that means they could be < 0 or > 0, I can see if we take a limit as x -> +- inf that the value of x tends to 1 as seen on the graph by the asymptote, I think its something to do with this asymptote but I’m unsure how to construct this.
Any help is greatly appreciated, many thanks
Original post by BrandonS15
Hello,
I’m currently having trouble in attempting this question:
095E6EA9-19F4-49A5-8905-234D57E3EA6C.jpg.jpeg

I’m completely confused as to where to begin in proving part (i) I have tried to take a general graph of y = 1 + (1/x) and sketching it to help me but its of no use as far as I can see, I know from the question that x1, x2, x3, ... , xn cannot equal 0 but that means they could be < 0 or > 0, I can see if we take a limit as x -> +- inf that the value of x tends to 1 as seen on the graph by the asymptote, I think its something to do with this asymptote but I’m unsure how to construct this.
Any help is greatly appreciated, many thanks


Hint:

Just look at the first equation - definition of x1.

Recall x1,x2,...,xn are positive.
Reply 2
Original post by ghostwalker
Hint:

Just look at the first equation - definition of x1.

Recall x1,x2,...,xn are positive.


Okay I understand that they are all positive so they are above 0 (combining the division by zero constraint and the positive given fact) so I’ve done this:
996C4A52-D509-42A4-8341-A1ADE00C67EC.jpg.jpeg

But I’m not sure if my argument is mathematically rigorous enough to be considered as a valid answer, by looking at the definition for each term I can see it works in a cyclic manner and that if any term is above 0 then adding a positive non zero number to 1 makes any term above 1 as described but how do I write this formally using the given equations?
Original post by BrandonS15
Okay I understand that they are all positive so they are above 0 (combining the division by zero constraint and the positive given fact) so I’ve done this:
996C4A52-D509-42A4-8341-A1ADE00C67EC.jpg.jpeg

But I’m not sure if my argument is mathematically rigorous enough to be considered as a valid answer, by looking at the definition for each term I can see it works in a cyclic manner and that if any term is above 0 then adding a positive non zero number to 1 makes any term above 1 as described but how do I write this formally using the given equations?



Seems to be a lot of superfluous lines there, and it's difficult to follow any reasoning.

I'd just say.

For all i,

xi>0x_i>0

Hence 1/xi>01/x_i >0

and 1+1/xi>11+1/x_i>1

Looking at the defintions of x_i, we have x1,...,xn>1x_1,...,x_n >1
(edited 3 years ago)
Reply 4
Original post by ghostwalker
Seems to be a lot of superfluous lines there, and it's difficult to follow any reasoning.

I'd just say.

For all i,

xi>0x_i>0

Hence 1/xi>01/x_i >0

and 1+1/xi>11+1/x_i>1

Looking at the defintions of x_i, we have x1,...,xn>1x_1,...,x_n >1


Okay that makes concise sense now, therefore part (ii) comes out easily, for part (iii) I have tried substitution to see if anything happens and its just putting me through a loop, I’m trying to prove specific cases and therefore use some form of inductive proof but I’m having no luck, I’m not sure where to go
Original post by BrandonS15
Okay that makes concise sense now, therefore part (ii) comes out easily, for part (iii) I have tried substitution to see if anything happens and its just putting me through a loop, I’m trying to prove specific cases and therefore use some form of inductive proof but I’m having no luck, I’m not sure where to go


This whole question is actually covered in Siklos's book, and it's probably more useful for you to use that rather than me just telling you the next step.

You have x1-x2 in terms of x3,x4, and basically just need to work down the line until you get to x1-x2 again.
Reply 6
Original post by ghostwalker
This whole question is actually covered in Siklos's book, and it's probably more useful for you to use that rather than me just telling you the next step.

You have x1-x2 in terms of x3,x4, and basically just need to work down the line until you get to x1-x2 again.


Reading through his solution, I’m still not quite sure how the result is proved, it appears he has assumed the result to prove the final part which makes sense but in proving the third part is where I’m confused, currently I’ve gotten this:
FA7488B2-01A7-45C7-B166-4316F9A00486.jpg.jpeg
I can backwards substitute but it doesn’t seem to make me achieve anything, how do I combine the cyclic nature of the equations and this idea of using part (ii) to move forward?
Sorry for the questions, I’m struggling to get my head around this
(edited 3 years ago)
Reply 7
Original post by ghostwalker
This whole question is actually covered in Siklos's book, and it's probably more useful for you to use that rather than me just telling you the next step.

You have x1-x2 in terms of x3,x4, and basically just need to work down the line until you get to x1-x2 again.


922E9D09-47F0-4CE6-A12C-C64886B40E18.jpg.jpeg
So letting n= 5 I have gotten to this expression at the bottom to which I’m not sure where its leading, how does this help me show that they (x1 x2 x3 ... xn) are equal?
Original post by BrandonS15
922E9D09-47F0-4CE6-A12C-C64886B40E18.jpg.jpeg
So letting n= 5 I have gotten to this expression at the bottom to which I’m not sure where its leading, how does this help me show that they (x1 x2 x3 ... xn) are equal?


You need one step further, substituting for x5x1x_5-x_1 to complete the "circle".

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