Thursday, January 4, 2018

Fun with Triangle and Square Numbers

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!)

1 comment:

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