Solution:

Here's an elegant way:

1 + 30 = 31
2 + 29 = 31
3 + 28 = 31
.....................
.....................
15 + 16 = 31

Adding the above equations gives

1 + 2 + 3 + ....... + 30 = 15 × 31 = 465

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?