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August 25th, 2013, 03:08 AM  #1 
Senior Member Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2  Algebra trick to make this induction proof work?
Hello there, For two days, I have tried to prove that this is true using induction: is true. I checked. The best I could ever arrive at, assuming to be true, was: . However, I suspect there is a prettier solution, and I am unable to continue my happy life without knowing what it was. Are any of you able to work it out? Thank you for your time. Kind regards, Marius 
August 25th, 2013, 05:45 AM  #2 
Senior Member Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2  Re: Algebra trick to make this induction proof work?
I found the solution myself: Thank you for your time. Kind regards, Marius 
August 25th, 2013, 05:52 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,641 Thanks: 2083 
Your work is okay. Obviously, ?(k+2)/?(k+1) + 1 > 2. Multiplying by ?(k+2)/?(k+1)  1 gives 1/(k+1) > 2(?k+2)/?(k+1)  1), which implies 1 > 2(?(k+2)?k+1))  2(k + 1), as needed to complete the induction proof. 
August 25th, 2013, 05:56 AM  #4 
Senior Member Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2  Re: Algebra trick to make this induction proof work?
Thanks skipjack, what do you think of my other attempt? Thanks.M 
August 25th, 2013, 07:08 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,641 Thanks: 2083 
Both are okay. The first is abbreviated and unfinished, whilst the second fills out the first and finishes it.


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