Talk:Square pyramidal number: Difference between revisions

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== Quadrature of the Parabola with the "square pyramidal number" (new proof) ==

I found that the "square pyramidal number" can be used to prove the theorem of Archimedes on the area of ​​parabolic segment. The proof, carried out without the use of "mathematical analysis", is on the site "Leggendo Archimede" on page 02, at the following web address:

::https://sites.google.com/site/leggendoarchimede/

https:

[[File:ARCH01.gif|thumb|Quadrature of the parabola]]

:https://sites.google.com/site/leggendoarchimede/

Below we show a summary of the proof.

:On page 07 of that site, in the article "Numeri e geometria", is exposed an ingenious method to derive the "square pyramidal number", and others sums of the same type, using geometric patterns in the spaces of 2 and 3 dimensions. Luciano Ancora - 11:22, 5 mag 2013 <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]]) </span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

Proposition: ''The area of ​​ parabolic segment is a third of the triangle ABC''.

Divide AB and BC into 6 equal parts and use the green triangle as measurement unit of the areas.

The triangle ABC contains 6^{<small>3</small>} green triangles.

The parabola circumscribed figure (in red) contains:

:A(cir.) = 6^{<small>.</small>}1 + 5^{<small>.</small>}3 + 4^{<small>.</small>}5 + 3^{<small>.</small>}7 + 2^{<small>.</small>}9 + 1^{<small>.</small>}11 = 91 green triangles. (3)

The sum (3) can be written:

:A(cir.) = 6 + 11 + 15 + 18 + 20 + 21 , that is:

:6+

:6+5+

:6+5+4+

:6+5+4+3+

:6+5+4+3+2

:6+5+4+3+2+1

or rather:

:A(cir.) = sum of the squares of first 6 natural numbers !

Generally, for any number n of divisions of AB and BC, it is:

# The triangle ABC contains n^{<small>3</small>} green triangles

# A_n(cir.) = sum of the squares of first n natural numbers

So, the saw-tooth figure that circumscribes the parabolic segment can be expressed with the "square pyramidal number" of number theory!

For the principle of mathematical induction, this circumstance (which was well hidden in (3)) we can reduce the proof to the simple check of the following statement:

: the sequence 1, 5/8, 14/27, 30/64, ....., P_n/n³, ..... tends at number 1/3, as n tends to infinity (4a)

where the numerator of the sequence terms is the nth square pyramidal number P_n.

But (4a) state that: the area (measured in green triangles) of the circumscribed figure is one-third the area of ​​the triangle ABC, at the limit of n = infinity.

:::End of proof