# Simplify 1/15+1/35+1/63+1/99+1/143?

Oct 27, 2017

$\frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} + \frac{1}{143} = \frac{5}{39}$

#### Explanation:

$\frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} + \frac{1}{143}$

= $\frac{1}{3 \times 5} + \frac{1}{5 \times 7} + \frac{1}{7 \times 9} + \frac{1}{9 \times 11} + \frac{1}{11 \times 13}$

= $\frac{7 \times 9 \times 11 \times 13 + 3 \times 9 \times 11 \times 13 + 3 \times 5 \times 11 \times 13 + 3 \times 5 \times 7 \times 13 + 3 \times 5 \times 7 \times 9}{3 \times 5 \times 7 \times 9 \times 11 \times 13}$

= $\frac{9009 + 3861 + 2145 + 1365 + 945}{135135}$

= $\frac{17325}{135135}$

= $\frac{3465}{27027}$ - dividing by $5$

= $\frac{385}{3003}$ - dividing ny $9$

= $\frac{35}{273}$ - dividing by $11$

= $\frac{5}{39}$ - dividing by $7$

Oct 27, 2017

5/39

#### Explanation:

Given, $\frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} + \frac{1}{143}$

First of all multiply Numerator and Denumerator by 2/2 and we get,

$\Rightarrow \frac{2}{2} \left(\frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} + \frac{1}{143}\right)$

$\Rightarrow \frac{1}{2} \left(\frac{2}{15} + \frac{2}{35} + \frac{2}{63} + \frac{2}{99} + \frac{2}{143}\right)$

$\Rightarrow \frac{1}{2} \left[\left(\frac{1}{3} - \frac{1}{5}\right) + \left(\frac{1}{5} - \frac{1}{7}\right) + \left(\frac{1}{7} - \frac{1}{9}\right) + \left(\frac{1}{9} - \frac{1}{11}\right) + \left(\frac{1}{11} - \frac{1}{13}\right)\right]$

$\Rightarrow \frac{1}{2} \left[\frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} + \frac{1}{9} - \frac{1}{11} + \frac{1}{11} - \frac{1}{13}\right]$

$\Rightarrow \frac{1}{2} \left[\frac{1}{3} - \frac{1}{13}\right]$

$\Rightarrow \frac{1}{2} \left[\frac{13 - 3}{39}\right]$

$\Rightarrow \frac{1}{2.} \frac{10}{39}$

$\Rightarrow \frac{5}{39}$