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[[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 18:22, 31 August 2009 (UTC)
== Sum of the first "n" cubes - even cubes - odd cubes (geometrical proofs) ==
[[File:Idea01.gif|thumb|The "IDEA"]]
[[File:Sum Cubes.gif|thumb|New method for summing the first "n" cubes]]
[[File:Evencubes.gif|thumb|Sum of the first "n" even cubes]]
[[File:Oddcubes.gif|thumb|Sum of the first "n" odd cubes]]
Starting from the basic idea described in the first animation, we introduced a new procedure to obtain formulas for summing the first "n" cubes, even cubes and odd cubes. This method, called "Successive Transformations Method", consists in an inductive handling of a geometric model, in order to obtain another equivalent which gives evidence of the searching formulas.
See the animations.
Consider the final transformation that you see in the second animation, and we compare this figure with the square base parallelepiped that contains it. We expect that the ratio between these figures becomes 1/2 to infinity. Performing calculations with Excel, you see that this is true. In addition, as has happened in the discussion at the link http://upload.wikimedia.org/wikipedia/commons/6/6c/Pubblicazione_english.pdf, you encounter these other amazing results:
<sub>n</sub>
lim (Σ<sub>n</sub> n<sup>3</sup>)/(Σ<sub>n</sub> n). n<sup>2</sup> = 1/2
<sup>n→∞ </sup><sup>1</sup>
<sub>n</sub>
lim (Σ<sub>n</sub> n<sup>5</sup>)/(Σ<sub>n</sub> n). n<sup>4</sup> = 1/3
<sup>n→∞ </sup><sup>1</sup>
<sub>n</sub>
lim (Σ<sub>n</sub> n<sup>7</sup>)/(Σ<sub>n</sub> n). n<sup>6</sup> = 1/4
<sup>n→∞ </sup><sup>1</sup>
<sub>n</sub>
lim (Σ<sub>n</sub> n<sup>9</sup>)/(Σ<sub>n</sub> n). n<sup>8</sup> = 1/5
<sup>n→∞ </sup><sup>1</sup>
which, by induction, can be generalized in a formula. Note that the denominators of the results are the positions of the exponents in the numerator in the sequence of odd numbers.
The induction principle enshrines the validity of this "theorem". Its algebraic proof would be an exciting challenge between the insiders.--[[Special:Contributions/79.17.53.156|79.17.53.156]] ([[User talk:79.17.53.156|talk]]) 09:21, 3 August 2013 (UTC)
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