Triangular Number

Carl Fredrich Gauss, a German mathematician and physicist, discovered in 1796 that every positive integer may be expressed as the sum of three triangular numbers.

The depiction of numbers in the form of an equilateral triangle arranged in a series or sequence is known as the triangular number sequence. The numerals are arranged in the following order: 1, 3, 6, 10, 15, 21, 28, 36, 45, and so on. Dots represent the numbers in the triangular design. The sequence of triangular numbers is formed by adding the previous number and the order of the next number.

Meaning of triangular numbers :-

Numbers that are triangular Tn are figurate numbers that can be organised in several rows of items with each row containing one more element than the previous row.

0 , 1 , 3 , 6 , 10 , 15 , 21 , 28 , 36 , 45 , 55 , 66 , 78 , 91 , 105 , 120 , 136 , 153 , 171 , 190 , 210 , 231 , 253 , 276 , 300 , 325 , 351 , 378 , 406 , 435 , 465 , 496 , 528 , 561 , 595 , 630 , 666 , 703 , 741 , 780 , 820 , 861 , 903 , 946 , 990 , 1035 , 1081 , 1128 , 1176 , 1225 , 1275 , 1326 , 1378 , 1431 , …… are the some of the sequences of the triangular numbers.

Sum of triangular numbers :-

You’ll see that the next number in the sequence is added with an extra row in the pattern of triangular numbers. Let us go over it in more depth.

• The first number is one.
• To the first number, a row with two dots is added in number 2.
• In number 3, three dots are added to the second number to make a row.
• In number 4, a row of four dots is added to the third number, and so on.

As a result, the series created here follows the pattern:

1 , 1 + 2 , 1 + 2 + 3 , ……..etc

How to find triangular numbers :-

Triangular numbers are things found in an equilateral triangle (also referred to as triangle numbers). N is the number of black dots in a triangular pattern with n black dots on each side, and is equal to the sum of all “n” natural integers from “1” to “n.” Starting with the 0th triangular number, an arrangement of triangular numbers is as follows: 

0 , 1 , 3 , 6 , 10 , 15 , 21 , 28 , 36 , 45 , 55 , 66 , 78 , 91 , 105 , 120 , 136 , 153 , 171 , 190 , 210 , 231 , 253 , 276 , 300 , 325 , 351 , 378 , 406 , 435 , 465 , 496 , 528 , 561 , 595 , 630 , 666 ….etc .

The formulas for finding the triangle numbers are :-

Tn = 1 + 2 + 3 + 4 + …….. + n = (n (n + 1))/2

Where (n+1)/2 is known as the binomial coefficient .It denotes the number of distinct pairs from which N+1 items can be chosen. 

We can state that the sum of n natural numbers equals a triangular number, or that the summation of natural numbers equals a triangular number, using the formula above. A square number is always the result of adding two consecutive natural integers.  

Relationships between triangular numbers and other figurate numbers are numerous.

Simply said, the sum of two consecutive triangular numbers is a square number, with the sum equaling the square of the difference between the two values (and thus the difference of the two being the square root of the sum).

Conclusion :-

A triangular number, sometimes known as a triangle number, is a number that counts things that are organised in an equilateral triangle. Square numbers and cube numbers are instances of figurate numbers, as are triangular numbers. The nth triangular number is equal to the sum of the n natural numbers from 1 to n and equals the number of dots in a triangle arrangement with n dots on each side. Starting with the 0th triangular number, the sequence of triangular numbers .