Triangular number

A Triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The nth triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The first 100 terms sequence of triangular numbers, starting with the 0th triangular number, are

(sequence A000217 in the OEIS)

wikipedia/en/Triangular%20number

Triangular numbers are numbers that can be arranged into an equilateral triangle, like the dots in bowling pins. The sequence starts with 1, and each subsequent number is found by adding the next consecutive integer (

1,1+2=3,1+2+3=6,1+2+3+4=101 comma 1 plus 2 equals 3 comma 1 plus 2 plus 3 equals 6 comma 1 plus 2 plus 3 plus 4 equals 10

1,1+2=3,1+2+3=6,1+2+3+4=10

, and so on). The

nthn raised to the t h power

𝑛𝑡ℎ

triangular number can be calculated using the formula

Tn=n(n+1)2cap T sub n equals the fraction with numerator n open paren n plus 1 close paren and denominator 2 end-fraction

𝑇𝑛=𝑛(𝑛+1)2

.
 

Examples

• 1st: 1 dot

• 2nd: 3 dots (

  

  1+21 plus 2

  1+2

  )

• 3rd: 6 dots (

  

  1+2+31 plus 2 plus 3

  1+2+3

  )

• 4th: 10 dots (

  

  1+2+3+41 plus 2 plus 3 plus 4

  1+2+3+4

  )

• 5th: 15 dots (

  

  1+2+3+4+51 plus 2 plus 3 plus 4 plus 5

  1+2+3+4+5

  ) 

• Triangular Number Sequence - Math is Fun

  This is the Triangular Number Sequence: 1, 3, 6, 10, 15, 21, 28, 36, 45, … It is simply the number of dots in each triangular pa…

  Math is Fun

  

How to find a triangular number

1. Using the sum of consecutive integers:

   To find the

   

   nthn raised to the t h power

   𝑛𝑡ℎ

   triangular number, simply add all the whole numbers from 1 to

   

   nn

   𝑛

   . For example, the

   

   5th5 raised to the t h power

   5𝑡ℎ

   triangular number is

   

   1+2+3+4+5=151 plus 2 plus 3 plus 4 plus 5 equals 15

   1+2+3+4+5=15

   . 

2. Using the formula:

   For larger numbers, the formula

   

   Tn=n(n+1)2cap T sub n equals the fraction with numerator n open paren n plus 1 close paren and denominator 2 end-fraction

   𝑇𝑛=𝑛(𝑛+1)2

   is much faster. 

   • To find the

     

     100th100 raised to the t h power

     100𝑡ℎ

     triangular number, use

     

     n=100n equals 100

     𝑛=100

     :

     

     T100=100(100+1)2=100×1012=5050cap T sub 100 equals the fraction with numerator 100 open paren 100 plus 1 close paren and denominator 2 end-fraction equals the fraction with numerator 100 cross 101 and denominator 2 end-fraction equals 5050

     𝑇100=100(100+1)2=100×1012=5050

     .