Talk:Square pyramidal number: Difference between revisions

Revision as of 06:11, 22 May 2013

Revision as of 06:13, 22 May 2013

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[[File:Pyram.jpg|thumb|tridimensional model]]

We represent the pyramid number P₆ = 91 with cubes of unit volume, as shown, and inscribe in building a pyramid (in red). Let V₆ the volume of the inscribed pyramid. To obtain P₆ you may add to V₆ the excess external volume to the red pyramid. Such excess is: 2/3 for each cube placed on the central edge, and 1/2 for the cubes forming the steps of the building (enlarge for a better look of highlighted part). Then, calculating one has:

:P₆ = V₆+(2/3)*6+(1+2+3+4+5)

: For the induction principle, will be:

:P_n = V_n+(2n)/3+Σ(from 1 to n-1)n

: For the induction principle, will be:

:P_n = n³/3+2n/3+(n²+n)/2-n

:P_n = (2n³+3n²+n)/6

:P_n = V_n+(2n)/3+Σ(from 1 to n-1)n

:P_n = n³/3+2n/3+(n²+n)/2-n

:P_n = (2n³+3n²+n)/6