$\frac{1}{3}(\frac{1}{{1 + 2}} + \frac{1}{{1 + 2 + 3}} + \frac{1}{{1 + 2 + 3 + 4}} + ... + \frac{1}{{1 + 2 + 3 + ... + 29}}) = \frac{1}{3}(\frac{1}{{\frac{{2 \times 3}}{2}}} + \frac{1}{{\frac{{3 \times 4}}{2}}} + \frac{1}{{\frac{{4 \times 5}}{2}}} + ... + \frac{1}{{\frac{{29 \times 30}}{2}}}) = $
$\frac{1}{3}(\frac{2}{{2 \times 3}} + \frac{2}{{3 \times 4}} + \frac{2}{{4 \times 5}} + ... + \frac{2}{{29 \times 30}}) = \frac{2}{3}(\frac{1}{{2 \times 3}} + \frac{1}{{3 \times 4}} + \frac{1}{{4 \times 5}} + ... + \frac{1}{{29 \times 30}})$
$\frac{2}{3}(\frac{1}{2} - \frac{1}{{30}}) = \frac{2}{3}(\frac{{14}}{{30}}) = \frac{{14}}{{45}}$