# What is the relationship between triangular numbers and natural number?

## What is the relationship between triangular numbers and natural number?

If you observe, the difference between consecutive triangular numbers results in a natural number series 1, 2, 3, 4, 5, 6, and so on. That is, the nth triangular number is obtained by adding “n” to the (n-1)th triangular number. Ex: 5th Triangular Number = 5 + 4th Triangular Number = 5 + 10 = 15.

**What is first triangular number?**

The sequence of triangular numbers, starting with the first triangular number, is. 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…

### How to find the sum of first n triangular numbers?

n t h triangular number is the sum of n consecutive natural numbers from starting which is simply n ( n + 1) / 2. You want sum of first n triangular numbers. Just take the sum Σ i = 1 n i ( i + 1) 2. Proof: we can prove it in an inductive way. Let n=k.

**What is the sum of the first n natural numbers?**

By definition, a triangular number is the sum of the first n natural numbers. So, 3 is a triangular number because 3 = 1 + 2, and similarly, 28 is a triangular number because 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7.

## What is the next triangular number from 1 to 3?

We begin with 1: We say that 1 is the first triangular number. To form the next, we add 2: So the next triangular number is 3. The number we add to the previous triangular number is called the gnomon (NOH-mon). We added the gnomon 2 to 1.

**How do you find the sum of the k-th triangular numbers?**

Since the k -th triangular number is T(k) = k ( k + 1) 2, so your sum is n ∑ k = 1k(k + 1) 2 = 1 2( n ∑ k = 1k2 + n ∑ k = 1k) The second summation is T(n), the first summation is 1 3n(n + 1 2)(n + 1) (a nice way to memorize it), you find it in several places (the book “Concrete Mathematics” by Graham,…