Solution:

Here's an elegant way:

1 + 34 = 35
2 + 33 = 35
3 + 32 = 35
.....................
.....................
17 + 18 = 35

Adding the above equations gives

1 + 2 + 3 + ....... + 34 = 17 × 35 = 595

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?