Tính tổng biểu thức: 1/3 + 1/15 + 1/35 + 1/63 + 1/99 + 1/143 + 1/195

Đặt \[A = \frac{1}{3} + \frac{1}{{15}} + \frac{1}{{35}} + \frac{1}{{63}} + \frac{1}{{99}} + \frac{1}{{143}} + \frac{1}{{195}}\]

\[2A = \frac{2}{3} + \frac{2}{{15}} + \frac{2}{{35}} + \frac{2}{{63}} + \frac{2}{{99}} + \frac{2}{{143}} + \frac{2}{{195}}\]

\[ = \frac{2}{{1 \cdot 3}} + \frac{2}{{3 \cdot 5}} + \frac{2}{{5 \cdot 7}} + \frac{2}{{7 \cdot 9}} + \frac{2}{{9 \cdot 11}} + \frac{2}{{11 \cdot 13}} + \frac{2}{{13 \cdot 15}}\]

\[ = \frac{1}{1} - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} + \frac{1}{9} - \frac{1}{{11}} + \frac{1}{{11}} - \frac{1}{{13}} + \frac{1}{{13}} - \frac{1}{{15}}\]

\[ = 1 - \frac{1}{{15}} = \frac{{14}}{{15}}\]

Suy ra \[A = \frac{{14}}{{15}}:2 = \frac{7}{{15}}.\]