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Two arithmetic series

  1. Oct 18, 2011 #1
    1. The problem statement, all variables and given/known data

    When x=! -1, y=! -1, x=! -y then x and y are two numbers so that 1/(x+1) + 1/(x+y) + 1/(y+1)... is an arithmetic serie. Show that then also x2 + 1 + y2.... must be an arithmetic serie.

    3. The attempt at a solution

    I tried to find the differentials between each number in the first line, then make them equal each other but I really didn't get anywhere. I figured out though that if x=y=1 then both of the series are arithmetic.

    Anyways can I get some help pls?
     
  2. jcsd
  3. Oct 18, 2011 #2

    Dick

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    a+b+c+.... an arithmetic series if b-a=c-b, right? Apply that to both your series and show they lead to the same condition on x and y.
     
  4. Oct 18, 2011 #3
    Yes, I wrote in the OP that I tried that but I didn't get any smarter.. can you maybe please solve this problem and show me how you did it?

    BTW: What do you mean by condition? You mean that x is the same number in both the series, and y is the same number in both the series? I got y=0.216 and Y=1, and x=0.37 and x=1, if I remember correct. That was wrong.
     
  5. Oct 18, 2011 #4

    Dick

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    Look, if x^2+1+y^2+... is arithmetic then 1-x^2=y^2-1. Yes? That leads to the 'condition' that 2=x^2+y^2. There are lots of solutions to that. Like x=0 and y=sqrt(2). All you have to do is do some algebra to show that the other series gives you the same equation in x and y.
     
  6. Oct 18, 2011 #5
    Well sure x=y=1 is a solution to both but that's not showing anything is it? I already did that difference between each joint stuff a while ago, and I got

    2=y^2 + x^2 in the second series and
    2= (1/y) + (1/x) in the first.

    that doesn't really tell me anything.. And are you suggesting that the difference in the first serie and the second are the same? Whut?
     
    Last edited: Oct 18, 2011
  7. Oct 18, 2011 #6

    Dick

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    No, the differences aren't the same between the two series. What is true is that b-a=c-b for each series. I think you did the algebra wrong for the second series. Can you show how you got 2= (1/y) + (1/x)?
     
  8. Oct 18, 2011 #7
    b-a= (1/y) - 1, c-b= 1 - 1/x. c-b=b-a. (1/y) - 1 = 1 - 1/x => 2 = 1/y + 1/x.

    But I don't get it, what am I supposed to do with this? can somebody help? How do I prove that as a result of the first serie, the second one must also be arithmetic??
     
  9. Oct 18, 2011 #8

    Dick

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    You can't simplify 1/(x+y)-1/(x+1) to 1/y-1 by magically canceling the x. That's wrong algebra. You need to find a common denominator if you are going to combine them.
     
  10. Oct 18, 2011 #9
    ah damn, you're right, I've been doing lots of mistakes of this type lately (sleep deprivation I guess).

    OK that was the root of the entire problem, I mixed up the fractions-math. Sorry. thanks for the help, I get it now.
     
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