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DOUBELL
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 MATHEMATICS   « on: Sep 1st, 2011, 8:45am » Quote Modify

CAN SOMEONE PROVE BY Mathematical induction that (2r)^3 = 2 (n^2) (n+1)^2 . need help with this one.
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towr
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 Re: MATHEMATICS   « Reply #1 on: Sep 1st, 2011, 8:52am » Quote Modify

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.
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DOUBELL
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 Re: MATHEMATICS   « Reply #2 on: Sep 1st, 2011, 9:11am » Quote Modify

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.
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pex
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 Re: MATHEMATICS   « Reply #3 on: Sep 1st, 2011, 11:41am » Quote Modify

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).
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ThudnBlunder
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 Re: MATHEMATICS   « Reply #4 on: Sep 1st, 2011, 12:22pm » Quote Modify

Maybe LHS should be (2n3)
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pex
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 Re: MATHEMATICS   « Reply #5 on: Sep 1st, 2011, 12:31pm » Quote Modify

on Sep 1st, 2011, 12:22pm, ThudnBlunder wrote:
 Maybe LHS should be (2n3)

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ThudnBlunder
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 Re: MATHEMATICS   « Reply #6 on: Sep 1st, 2011, 12:42pm » Quote Modify

on Sep 1st, 2011, 12:31pm, pex wrote:

Thank you, pex. [fingernail_polishing_smiley]
 « Last Edit: Sep 1st, 2011, 4:01pm by ThudnBlunder » IP Logged

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towr
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 Re: MATHEMATICS   « Reply #7 on: Sep 1st, 2011, 12:52pm » Quote Modify

Ah, then it makes sense

base case,
0..0 (2r)3 = 0 = 2 * 02 * (0+1)2

induction under assumption it's true for every natural number smaller than n:
0..n (2r)3
= {move last term from sum}
0..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
 « Last Edit: Sep 1st, 2011, 12:54pm by towr » IP Logged

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DOUBELL
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 Re: MATHEMATICS   « Reply #8 on: Sep 1st, 2011, 1:52pm » Quote Modify

on Sep 1st, 2011, 12:22pm, ThudnBlunder wrote:
 Maybe LHS should be (2n3)

THAT IS IN FACT CORRECT ABOUT THE LEFT HAND SIDE
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Michael Dagg
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 Re: MATHEMATICS   « Reply #9 on: Feb 29th, 2012, 9:53pm » Quote Modify

Gee. One might ask if induction is valid within an
induction argument itself. What you do think?
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Michael Dagg
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 Re: MATHEMATICS   « Reply #10 on: Oct 29th, 2012, 11:13am » Quote Modify

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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