## Triangular NumbersA triangular number is also called spread out in a series or sequence. Others also include Cube numbers and square numbers.
Examples of Triangular - Aeroplanes flying together constitutes this formation.
- Flocks of birds fly in the triangular formation.
Examples of Triangular number includes ## The Formula for the Triangular NumberThe formula for triangular numbers is given below: The mathematical induction can prove this formula Now assume that for some natural number q Proving the formula using mathematical induction Let's assume that some natural number n Adding (n + 1) to both sides of the equation From here we can conclude that it is true for k = 1, n and for (n + 1) Where (n + 1)/ 2 is termed a binomial coefficient. It shows the number of distinct pairs that can be chosen from n + 1 objects, and it can be expressed as (n + 1) Factorial / (n + 1- 2) factorial 2 factorial It has been simplified as {n (n + 1)}/2. So, we can conclude from the above that the sum of n natural numbers gives a triangular number. We can infer that the summation of the natural number gives a triangular number. ## Properties of the Triangular numbers- The sum of two consecutive triangular numbers gives a square number.
Suppose = 3 + 6 = 9 = 3 x 3 - If A is a triangular number, 9 * A + 1 will also be a Triangular number.
9 * A+ 1 = 9 x 6 + 1 = 55 9 * A + 1 = 9 x 10 + 1 = 91 - 2, 4, 7, or 9 cannot came at the end of triangular number.
- If A is a triangular number, then 8 * A + 1 will always be a perfect square
8 * A + 1 = 8 * 3 + 1= 24 + 1 = 25 = 5 x 5 8 * A + 1 = 8 * 6 + 1 = 48 + 1 = 49 = 7 x 7 - The Addition or sum of n consecutive cubes from 1 is equivalent to the square of the nth triangular number.
- Four specific triangular number in AP (Arithmetic Progression) doesn't exist.
- The sum or Addition of the squares of two consecutive triangular numbers gives a triangular number.
A A A ## Some Interesting Facts Related to Triangular Numbers- All perfect numbers also come in the category of a triangular number
- Sequence numbers 1, 11, 111, 1111, 11111, …… are all triangular numbers in base 9.
- 3 is the only triangular number that is prime.
## Palindromic Triangular numberThese numbers can be read the same forward as well as backward. For example, 55, 66, ## Reversible Triangular NumberThere are some triangular numbers when their reversal also results in the triangular number. Examples of reversible triangular numbers include 190, 171,153, 120, 820, 15051, and 17578. ## Square Triangular NumberAn infinite triangular number exists in the series, and the series gives squares. An example includes 1, 36, 1225, 41616…. ## Applications of Triangular NumberA triangular number is mostly applied in the Handshake problem, and it is one of the most prominent applications. ## Solved Example
Difference = 55 - 45 = 10. To determination of the difference for the next term, we need to add one more to the difference, and it will come as = 10 + 1 = 11 Hence the next term 55 + 11 = 66.
For example, we want to determine the 4 Next Topic1 Million in Crores |