Difference between revisions of "Triangular numbers"

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(Properties)
 
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[[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)=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