Question

What is the value of the following expression
$\left(\frac{1}{(2^2-1)}\right)+\left(\frac{1}{(4^2-1)}\right)+\left(\frac{1}{(6^2-1)}\right)+...+\left(\frac{1}{({20}^2-1)}\right)$ ?

• A. $\frac{9}{19}$
• B. $\frac{10}{19}$
• C. $\frac{10}{21}$
• D. $\frac{11}{21}$

Answer 1

Answer: (C)

$\frac{10}{21}$

$\Rightarrow \frac{1}{1\cdot 3}+\frac{1}{3\cdot 5}+\frac{1}{5\cdot 7}+....+\frac{1}{19\cdot 21}$
$\Rightarrow \frac{1}{2}(1-\frac{1}{3})+\frac{1}{2}(\frac{1}{3}-\frac{1}{5})+\frac{1}{2}(\frac{1}{5}-\frac{1}{7})+...+\frac{1}{2}(\frac{1}{19}-\frac{1}{21})$
$\Rightarrow \frac{1}{2}-\frac{1}{42}=\frac{(21-1)}{42}=\frac{20}{42}=\frac{10}{21}$