Math Proof

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Discus: SAT/ACT Tests and Test Preparation: July 2004 Archive: Math Proof
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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