The value of $$\frac{1}{{15}} + $$ $$\frac{1}{{35}} + $$ $$\frac{1}{{63}} + $$ $$\frac{1}{{99}} + $$ $$\frac{1}{{143}}$$ is :

$$ = \frac{1}{{15}} + \frac{1}{{35}} + \frac{1}{{63}} + \frac{1}{{99}} + \frac{1}{{143}}$$
$$ = \frac{1}{{3 \times 5}} + \frac{1}{{5 \times 7}} + \frac{1}{{7 \times 9}} + \frac{1}{{9 \times 11}} + \frac{1}{{11 \times 13}}$$
$$ = \frac{1}{2}\left[ {\frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} + \frac{1}{9} - \frac{1}{{11}} + \frac{1}{{11}} - \frac{1}{13}} \right]$$
$$ = \frac{1}{2}\left[ {\frac{1}{3} - \frac{1}{{13}}} \right]$$
$$ = \frac{1}{2} \times \frac{{10}}{{39}}$$
$$ = \frac{5}{{39}}$$