Solution:

Here's an elegant way:

1 + 50 = 51
2 + 49 = 51
3 + 48 = 51
.....................
.....................
25 + 26 = 51

Adding the above equations gives

1 + 2 + 3 + ....... + 50 = 25 × 51 = 1275

Food for thought:

The general formula is:

1 + 2 + 3 + ....... + n = n (n + 1) / 2.

Is the formula valid when n is an odd integer?