|By Abogachkov (Abogachkov) on Thursday, July 22, 2004 - 02:38 am: Edit|
can someone prove that a^3 + b^3 + c^3 = (a+b+c)^2? The constraints are that a, b, and c are in an arithmetic sequence (I think that is the constraint, but please check if that is true), and that the numbers are greater than 0.
|By Bulldogg (Bulldogg) on Thursday, July 22, 2004 - 03:38 am: Edit|
What do you mean by "a,b,c are in an arithmetic sequence"?
In general, even for consecutive numbers this isn't true...
|By Im_Blue (Im_Blue) on Thursday, July 22, 2004 - 04:21 am: Edit|
This isn't true in general. Counterexample: let a = 1, b = 3, c = 5. Then a^3 + b^3 + c^3 = 153, but (a + b + c)^2 = 81. Are you trying to find specific values of a, b, and c such that the conditions are met?
|By Feuler (Feuler) on Thursday, July 22, 2004 - 12:07 pm: Edit|
I believe the only sequences for which that is true are given by (x-sqrt(3x/2-x^2/2), x, x+ sqrt(3x/2-x^2/2) for 0 < x < 3. The only integer solution appears to be (1,2,3), and (0, 1, 2) if you relax the "all greater than 0" condition.
|By Tongos (Tongos) on Thursday, July 22, 2004 - 12:22 pm: Edit|
actually, the question is very valid, hes not saying that every arithmetic sequence satisfies the condition, he saying that this holds true for only arithmetic sequences.
|By Ubercollegeman (Ubercollegeman) on Thursday, July 22, 2004 - 01:12 pm: Edit|
I think he means that a^3 + b^3 + c^3 = (a+b+c)^2 implies that a, b, and c are in an arithmetic sequence.
You know, p->q. Either that or the problem doesn't make too much sense.
|By Optimizerdad (Optimizerdad) on Thursday, July 22, 2004 - 01:37 pm: Edit|
An alternative statement might be
'If a,b,c are in arithmetic sequence, and none of them equals zero, what are the conditions under which
a^3 + b^3 + c^3 = (a+b+c)^2 ?'
Feuler's neat proof shows the conditions under which this is true. (It starts, I think, by setting a=x-k, b=x, c=x+k, and then solving for k; you end up with k^2 = (3x - x^2)/2 ).
As he said, the only integer values of (a,b,c) which satisfy these conditions are (1,2,3). If non-integer values are allowed, then we have an infinite #solutions.
|By Tongos (Tongos) on Thursday, July 22, 2004 - 01:50 pm: Edit|
okay, well, now that we know that an arithmetic sequence exists to make this true. is there any numbers that can be plugged in for a,b and c (and to make the statement true) that don't have an arithmetic relation with eachother. If not, then the statement is true, that a, b and c have to follow an arithmetic path.
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