Difference between revisions of "Triangular number"

Revision as of 19:32, 27 January 2016

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

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

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

For example, the $n^{th}$ triangle number is $1 +2+3 + 4............. +(n-1)+(n)$

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.

Pascal's Triangle:

The rather simple recursive definition can be easily found by noting that $T_{n} = 1 + 2 + \ldots + (n-1) + n = (1 + 2 + \ldots + n-1) + n = T_{n-1} + n$ .

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

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-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>.
-Using the sum of an [[arithmetic series]] formula, a formula can be calculated for <math>\displaystyle T_n</math>:
+Using the sum of an [[arithmetic series]] formula, a formula can be calculated for <math>T_n</math>:
-:<math>T_n = \displaystyle\sum_{k=1}^{n}k = 1 + 2 + \ldots + n = \frac{n(n+1)}2</math>
+:<math>T_n =\sum_{k=1}^{n}k = 1 + 2 + \ldots + 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 + \ldots + (n-1) + n = (1 + 2 + \ldots + n-1) + n = T_{n-1} + n</math>.
+For example, the <math>n^{th}</math> triangle number is <math>1 +2+3 + 4............. +(n-1)+(n)</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.
+Pascal's Triangle:
+[[File: Pascal's Triangle.png]]
+The rather simple recursive definition can be easily found by noting that <math>T_{n} = 1 + 2 + \ldots + (n-1) + n = (1 + 2 + \ldots + n-1) + n = T_{n-1} + n</math>.
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Difference between revisions of "Triangular number"

Revision as of 19:32, 27 January 2016