Derive the formula for finding triangular numbers.

A triangular number is a number that can be represented as a triangle with dots. The formula for finding the nth triangular number is given by:

Tn = n(n+1)/2

where Tn is the nth triangular number.

To derive this formula, consider a triangle with n rows. The first row has 1 dot, the second row has 2 dots, the third row has 3 dots, and so on.

The total number of dots in the triangle can be calculated by summing the number of dots in each row. This can be represented as:

Tn = 1 + 2 + 3 + ... + n

This sum can be simplified using the formula for the sum of an arithmetic series:

Tn = n(n+1)/2

Therefore, the formula for finding the nth triangular number is Tn = n(n+1)/2.

Step 1: Understand the Definition of Triangular Numbers

Triangular numbers represent triangular patterns of dots. The nth triangular number, denoted by T(n), is the sum of the first n positive integers.

Example:
- T(1) = 1 (1 dot)
- T(2) = 3 (1 + 2 dots)
- T(3) = 6 (1 + 2 + 3 dots)

Step 2: Create an Expression for Triangular Numbers

We can create an expression for T(n) by summing the first n positive integers:

```
T(n) = 1 + 2 + 3 + ... + n
```

Step 3: Use the Sum of an Arithmetic Series

The sum of an arithmetic series (a + a + ... + a) with n terms is given by:

```
Sum = n - (a + l) / 2
```

where a is the first term and l is the last term.

Step 4: Substitute into the Expression for Triangular Numbers

Substituting a = 1 and l = n into the above formula, we get:

```
T(n) = n - (1 + n) / 2
```

Step 5: Simplify the Expression

Expanding and simplifying, we get:

```
T(n) = n - n / 2 + n / 2
T(n) = (n^2 + n) / 2
```

Therefore, the formula for finding triangular numbers is:

```
T(n) = (n^2 + n) / 2
```