Triangle number

The first six triangular numbers are shown as well as their general term. In addition to the exposed denotation, a triangular number can be indicated, placing the corresponding triangle side in brackets. For example, 10 is the triangular number of side 4, that is, T(4)=10.

A triangular number counts objects arranged in an equilateral triangle. The nth triangular number is the number of points in the triangular arrangement with n points on one side, and is equal to the sum of the n natural numbers from 1 to n, with 1 being the first triangular number by convention. Triangular numbers, along with other figurative numbers, were studied by Pythagoras and the Pythagoreans, who considered 10 written in a triangular form sacred, and which they called Tetraktys.

Formal definition

Each triangular number T_n is defined by the following formula:

Tn=n(n+1)2{displaystyle T_{n}={begin{matrix}{frac {n(n+1)}{2}}{matrix}}{,}

Demonstration

Be:

Tn=n(n+1)2{displaystyle T_{n}={begin{matrix}{frac {n(n+1)}{2}}{matrix}}{,}

Tn− − 1=(n− − 1)(n− − 1+1)2=n(n− − 1)2{displaystyle T_{n-1}={begin{matrix}{frac {(n-1+1)}{2}}}{matrix}}{begin{matrix}{matrix}{frac {n-1}}{2}}}{matrix}}{,}}

adding:

Tn+Tn− − 1=n(n+1)2+n(n− − 1)2{displaystyle T_{n}+T_{n-1}={begin{matrix}{frac {n(n+1)}{2}}end{matrix}}{begin{matrix}{matrix}{frac {n(n-1)}{2}}{matrix}}{,}{

ie:

Tn+Tn− − 1=n2{displaystyle T_{n}+T_{n-1}=n^{2},}

proving what is proposed. We can check this with any two consecutive triangular numbers, for example, with T₃ = 6 and T₄ = 10.

Indeed,

T4+T3=10+6=16=42{displaystyle T_{4}+T_{3}=10+6=16=4^{2},} Figure 2 (below, left) is said square.

Sum of two equal triangular numbers: oblong number

This figure shows how the triangular number T₄ results the oblong number of (5·4) points.

The sum of two equal triangular numbers gives us an oblong number, which forms the figure of a rhomboid. Let's look at its general term:

Tn+Tn=2Tn=2n(n+1)2{displaystyle T_{n}+T_{n}=2T_{n}=2{begin{matrix}{frac {n(n+1)}{2}}end{matrix}}}{,}

2Tn=n(n+1){displaystyle 2T_{n}=n(n+1),}

which is the search expression.

Sum of the first triangular numbers

The sum of the first n triangular numbers is also known as the tetrahedral number, so the nth tetrahedral number is the sum of the first n triangular numbers. His expression is:

S=n(n+1)(n+2)6{displaystyle S={frac {n(n+1)}{6}}}}}

Gauss and his theorem

In 1796, the German mathematician and scientist Carl Friedrich Gauss discovered that every positive integer can be represented as the sum of a maximum of three triangular numbers, a fact that he described in his journal with the same word used by Archimedes in his famous discovery: "Eureka! num= Δ + Δ + Δ." Note that this theorem does not imply that the triangular numbers are different (as in the case of 20 = 10 + 10), nor does it imply that there must be a solution with three triangular numbers that are different from zero. This is a special case of Fermat's polygonal number theorem.

The largest triangular number that can be represented by the formula 2^k − 1 is 4095 (Ramanujan–Nagell equation).

Polish mathematician Wacław Sierpiński wondered if there would be four different triangular numbers in the geometric progression. The Czech mathematician Kazimierz Szymiczek inferred that this statement was false. Chinese mathematicians Jin-Hui Fang and Yong-Gao Chen, professors in the Department of Mathematics at Nanjing Normal University, People's Republic of China, demonstrated this inference in 2007.