Difference between revisions of "Triangular number"
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− | The '''triangular numbers''' are the numbers <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> | + | Using the sum of an [[arithmetic series]] formula, a formula can be calculated for <math>T_n</math>: |
− | :<math>T_n = | + | :<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> | + | For example, the <math>n^{th}</math> triangle number is <math>1 +2+3 + 4............. +(n-1)+(n)</math> |
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+ | 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>. | ||
{{stub}} | {{stub}} |
Revision as of 20:32, 27 January 2016
The triangular numbers are the numbers which are the sum of the first
natural numbers from
to
.
Using the sum of an arithmetic series formula, a formula can be calculated for :
For example, the triangle number is
The formula for finding the triangular number can be written as
.
It can also be expressed as the sum of the row in Pascal's Triangle and all the rows above it. Keep in mind that the triangle starts at Row 0.
The rather simple recursive definition can be easily found by noting that .
This article is a stub. Help us out by expanding it.