Difference between revisions of "Triangular numbers"
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| − | 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) | + | [[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) | + | * {{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.
Graphical representation of the first six triangular numbers.
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
- V.E. Hoggatt, Jr and Marjorie Bicknell: Triangular numbers (1974)... (next) $(1.1)$
