# Difference between revisions of "Triangular number"

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− | 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>. | + | 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 n. | ||

+ | 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, 1+2, 1+2+3, ... etc. | ||

+ | giving the first few triangular numbers to be | ||

+ | <math>1, 3, 6, 10, 15, 21</math>. | ||

+ | |||

+ | A rather simple recursive definition can be found by noting that <math>T_{n} = 1 + 2 + \ldots + (n-1) + n = (1 + 2 + \ldots + n-1) + n = T_{n-1} + n</math>. | ||

+ | |||

+ | They are called triangular because you can make a triangle out of dots, and the number of dots will be a triangular number: | ||

+ | <asy> | ||

+ | int draw_triangle(pair start, int n) | ||

+ | { | ||

+ | real rowStart = start.x; | ||

+ | for (int row=1; row<=n; ++row) | ||

+ | { | ||

+ | for (real j=rowStart; j<(rowStart+row); ++j) | ||

+ | { | ||

+ | draw((j, start.y - row), linewidth(3)); | ||

+ | } | ||

+ | 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> | ||

+ | ==Formula== | ||

Using the sum of an [[arithmetic series]] formula, a formula can be calculated for <math>T_n</math>: | Using the sum of an [[arithmetic series]] formula, a formula can be calculated for <math>T_n</math>: | ||

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:<math>T_n =\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 formula for finding the <math>n^{th}</math> triangular number can be written as <math>\dfrac{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. | + | 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. |

+ | |||

− | |||

− | |||

− | |||

{{stub}} | {{stub}} |

## Latest revision as of 08:39, 7 July 2021

The **triangular numbers** are the numbers which are the sum of the first natural numbers from to .

## Definition

The triangular number is the sum of all natural numbers from one to n. That is, the triangle number is .

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

A rather simple recursive definition can be found by noting that .

They are called triangular because you can make a triangle out of dots, and the number of dots will be a triangular number:

## Formula

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

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.

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