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riddles >> putnam exam (pure math) >> MATHEMATICS
(Message started by: DOUBELL on Sep 1st, 2011, 8:45am)

Title: MATHEMATICS
Post by DOUBELL on Sep 1st, 2011, 8:45am
CAN SOMEONE PROVE BY Mathematical induction that (2r)^3 = 2 (n^2) (n+1)^2 . need help with this one.

Title: Re: MATHEMATICS
Post by towr on Sep 1st, 2011, 8:52am
I think you may have stated the problem incorrectly or incompletely. Since there seem to be no constraints on the values of r and n the two sides are plainly not equal for all values of n and r.

Title: Re: MATHEMATICS
Post by DOUBELL on Sep 1st, 2011, 9:11am
it is fact correct since for r=1  the LHs is (2)^3 = 8
AND FOR N =1 THE RIGHT HAND SIDE IS 2(1^2) (1+1)^2= 2 (2)^2 = 8.

Title: Re: MATHEMATICS
Post by pex on Sep 1st, 2011, 11:41am
I cannot find any other integer solutions than
r=0, n=-1
r=0, n=0
r=1, n=-2
r=1, n=1.
I don't see what mathematical induction could have to do with it, except perhaps in proving that there are no other solutions (or that there are, but I missed them).

Title: Re: MATHEMATICS
Post by ThudnBlunder on Sep 1st, 2011, 12:22pm
Maybe LHS should be http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/csigma.gif(2n3)

Title: Re: MATHEMATICS
Post by pex on Sep 1st, 2011, 12:31pm

on 09/01/11 at 12:22:53, ThudnBlunder wrote:
Maybe LHS should be http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/csigma.gif(2n3)

I am impressed by your mind-reading skills!

Title: Re: MATHEMATICS
Post by ThudnBlunder on Sep 1st, 2011, 12:42pm

on 09/01/11 at 12:31:09, pex wrote:
I am impressed by your mind-reading skills!

Thank you, pex. [fingernail_polishing_smiley]

Title: Re: MATHEMATICS
Post by towr on Sep 1st, 2011, 12:52pm
Ah, then it makes sense

[hide]base case,
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif0..0 (2r)3 = 0 = 2 * 02 * (0+1)2

induction under assumption it's true for every natural number smaller than n:
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif0..n (2r)3
= {move last term from sum}
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif0..n-1 (2r)3 + (2n)3
= {invoke induction hypothesis}
2 * n2 * (n-1)2 + 8n * n2
= {regroup terms}
2 * ((n-1)2 + 4n) * n2
= {simplify}
2 * n2 * (n+1)2[/hide]

Title: Re: MATHEMATICS
Post by DOUBELL on Sep 1st, 2011, 1:52pm

on 09/01/11 at 12:22:53, ThudnBlunder wrote:
Maybe LHS should be http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/csigma.gif(2n3)


THAT IS IN FACT CORRECT ABOUT THE LEFT HAND SIDE

Title: Re: MATHEMATICS
Post by Michael Dagg on Feb 29th, 2012, 9:53pm
Gee. One might ask if induction is valid within an
induction argument itself. What you do think?

Title: Re: MATHEMATICS
Post by Jack Hadin on Oct 29th, 2012, 11:13am
I think you have recorded the problem wrongly or perhaps incompletely. Because on a search engine appear to be no constraints throughout the principles of r along with n the two sides tend to be plainly not equal for every one of the principles of n also as r.



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