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August 25th, 2013, 03:08 AM   #1
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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
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August 25th, 2013, 05:45 AM   #2
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Re: Algebra trick to make this induction proof work?

I found the solution myself:

















Thank you for your time.

Kind regards,
Marius
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August 25th, 2013, 05:52 AM   #3
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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.
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August 25th, 2013, 05:56 AM   #4
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Re: Algebra trick to make this induction proof work?

Thanks skipjack, what do you think of my other attempt?

Thanks.M
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August 25th, 2013, 07:08 AM   #5
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Both are okay. The first is abbreviated and unfinished, whilst the second fills out the first and finishes it.
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