Talk:Square pyramidal number: Difference between revisions

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== Geometrical derivation of the formula ==

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

We represent the pyramid number P6P₆ = 91 with cubes of unit volume, as shown, and inscribe in building a pyramid (in red). Let V6V₆ the volume of the inscribed pyramid. To obtain P6P₆ you may add to V6V₆ 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

:P_n = n³/3+2n/3+(n²+n)/2-n

:P_n = (2n³+3n²+n)/6

:P_n = (2n³+3n²+n)/6 <small><span class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Ancora Luciano|Ancora Luciano]] ([[User talk:Ancora Luciano|talk]] • [[Special:Contributions/Ancora Luciano|contribs]]) 06:06, 22 May 2013 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->