Difference between revisions of "Triangular numbers"

From specialfunctionswiki
Jump to: navigation, search
(Properties)
 
(6 intermediate revisions by the same user not shown)
Line 1: Line 1:
The triangular numbers $T(n)$ are defined by the formula
+
The triangular numbers $T(n)$ are defined for $n=1,2,3,\ldots$ by the formula
 
$$T(n)=\displaystyle\sum_{k=1}^n k.$$
 
$$T(n)=\displaystyle\sum_{k=1}^n k.$$
 
They represent the number of ways to draw an equilateral triangle as in the image below.
 
They represent the number of ways to draw an equilateral triangle as in the image below.
Line 10: Line 10:
  
 
=Properties=
 
=Properties=
[[T(n) equals n(n+1)/2]]<br >
+
[[T(n)=n(n+1)/2]]<br >
 
[[T(n+1)=T(n)+n+1]]<br />
 
[[T(n+1)=T(n)+n+1]]<br />
 
[[n^2=T(n)+T(n-1)]]<br />
 
[[n^2=T(n)+T(n-1)]]<br />
[[T(n)^2=T(T(n))+T(T(n-1))]]<br />
+
[[T(n)^2=T(T(n))+T(T(n)-1)]]<br />
 +
[[T(n+1)^2-T(n)^2=(n+1)^3]]<br />
  
 
=References=
 
=References=
* {{PaperReference|Triangular numbers|1974|V.E. Hoggatt, Jr|author2=Marjorie Bicknell|next=T(n) equals n(n+1)/2}}
+
* {{PaperReference|Triangular numbers|1974|V.E. Hoggatt, Jr|author2=Marjorie Bicknell|next=T(n)=n(n+1)/2}} $(1.1)$
  
 
{{:Polygonal numbers footer}}
 
{{:Polygonal numbers footer}}
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 01:32, 30 May 2017

The triangular numbers $T(n)$ are defined for $n=1,2,3,\ldots$ by the formula $$T(n)=\displaystyle\sum_{k=1}^n k.$$ They represent the number of ways to draw an equilateral triangle as in the image below.

Properties

T(n)=n(n+1)/2
T(n+1)=T(n)+n+1
n^2=T(n)+T(n-1)
T(n)^2=T(T(n))+T(T(n)-1)
T(n+1)^2-T(n)^2=(n+1)^3

References

Polygonal numbers