Difference between revisions of "Triangular number"

Revision as of 19:46, 15 July 2020

The triangular numbers are the numbers $T_n$ which are the sum of the first $n$ natural numbers from $1$ to $n$ .

Definition

The $n^{th}$ triangular number is the sum of all natural numbers from one to $n$ . That is, the $n^{th}$ triangle number is $1 +2+3 + 4............. +(n-1)+(n)$ .

For example, the first few triangular numbers can be calculated by adding 1, 1+2, 1+2+3, ... etc.

   $}
   rowStart -= 0.5;
 }
 return 0;

}

for (int n=1; n<5; ++n) {

 real value= n*(n+1)/2;
 draw_triangle((value+5,n),n);
 label( (string) value, (value+5, -2));

} </asy>$ (Error compiling LaTeX. Unknown error_msg)

Formula

Using the sum of an arithmetic series formula, a formula can be calculated for $T_n$ :

$T_n =\sum_{k=1}^{n}k = 1 + 2 + \ldots + n = \frac{n(n+1)}2$

The formula for finding the $n^{th}$ triangular number can be written as $\dfrac{n(n+1)}{2}$ .

It can also be expressed as the sum of the $n^{th}$ row in Pascal's Triangle and all the rows above it. Keep in mind that the triangle starts at Row 0.

This article is a stub. Help us out by expanding it.

@@ Line 1: / Line 1: @@
-The '''triangular numbers''' are the numbers <math>\displaystyle T_n</math> which are the sum of the first <math>\displaystyle n</math> [[natural number]]s from <math>\displaystyle 1</math> to <math>\displaystyle n</math>.
+The '''triangular numbers''' are the numbers <math>T_n</math> which are the sum of the first <math>n</math> [[natural number]]s from <math>1</math> to <math>n</math>.
+==Definition==
+The <math>n^{th}</math> triangular number is the sum of all natural numbers from one to <math>n</math>.
+That is, the <math>n^{th}</math> triangle number is
+<math>1 +2+3 + 4............. +(n-1)+(n)</math>.
+For example, the first few triangular numbers can be calculated by adding
+, 1+2, 1+2+3, ... etc.
+    <math>}
+    rowStart -= 0.5;
+  }
+  return 0;
+}
+for (int n=1; n<5; ++n)
+{
+  real value= n*(n+1)/2;
+  draw_triangle((value+5,n),n);
+  label( (string) value, (value+5, -2));
+}
+</asy></math>
+==Formula==
+Using the sum of an [[arithmetic series]] formula, a formula can be calculated for <math>T_n</math>:
+:<math>T_n =\sum_{k=1}^{n}k = 1 + 2 + \ldots + n = \frac{n(n+1)}2</math>
+The formula for finding the <math>n^{th}</math> triangular number can be written as <math>\dfrac{n(n+1)}{2}</math>.
+It can also be expressed as the sum of the <math>n^{th}</math> row in [[Pascal's Triangle]] and all the rows above it. Keep in mind that the triangle starts at Row 0.
-Using the sum of an [[arithmetic series]] formula, a formula can be calculated for <math>\displaystyle T_n</math>:
-:<math>T_n = \displaystyle\sum_{i=1}n = 1 + 2 \cdots n = \frac{n(n+1)}2</math>
-The rather simple recursive definition can be easily found by noting that <math>\displaystyle T_{n} = 1 + 2 \cdots (n-1) + n = (1 + 2 \cdots n-1) + n = T_{n-1} + n</math>.
 {{stub}}

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Difference between revisions of "Triangular number"

Revision as of 19:46, 15 July 2020

Definition

Formula