Fun with Triangle and
Square Numbers
Formulas
Triangle
Numbers:
Tn
= COMB(n +1, 2) = (n^2 + n)/2
Square
Numbers:
Sn
= n^2
where
n is an integer, n ≥ 1
Mathematics between
Triangle and Square Numbers
2*Tn
2*Tn
= 2 * (n^2 + n)/2 = n^2 + n = Sn + n
Note
that 2*Tn is a rectangle number, which consists of a square with a row or
column attached to it. 2*Tn is an
integer.
3*Tn
3*Tn
= 3 * (n^2 + n)/2
We
can show that 3*Tn is an integer by showing that 3*Tn is an integer for both
even and odd n.
First
assume that n is even, let n = 2*k and k is a positive integer:
3
* Tn
=
3/2 * ((2*k)^2 + 2*k)
=
3/2 * (4*k^2 + 2*k)
=
3 * (2*k^2 + k)
The
result is an integer.
Next,
let n be an odd integer of the form of n = 2*k + 1. Then
3
* Tn
=
3/2 * ((2*k + 1)^2 + (2*k + 1))
=
3/2 * (4*k^2 + 4*k + 1 + 2*k + 1)
=
3/2 * (4*k^2 + 6*k + 2)
=
3 * (2*k^2 + 3*k + 1)
The
result is an integer.
Tn^2
Tn^2
= (n^2 + n)/2)^2
Expanding
Tn^2 to get:
=
(n^4 + 2*n^3 + n^2)/4
Tn^2
in terms of Sn and n:
Tn^2
=
(Sn^2 +2*n^3 + Sn)/4
=
(Sn^2 + 2*Sn*n + Sn)/4
=
(Sn^2 + Sn*(2*n + 1))/4
Clearly
Tn^2 is an integer since Tn is an integer, and multiplying two integers
generates another integer.
Square Numbers in terms
of Triangle Numbers
With
Sn = n^2 and Tn = (n^2 + n)/2
Tn
= (n^2 + n)/2
Tn
= (Sn + n)/2
2
* Tn = Sn + n
2
* Tn – n = Sn
Eddie
This
blog is property of Edward Shore, 2018.
(Happy New Year!)