Question

The value of
$S=\frac{1}{2\times 7}+\frac{1}{7\times 12}+\frac{1}{12\times 17}+\cdot \cdot \cdot +\frac{1}{225\times 257}$ is

• A. $\frac{5}{257}$
• B. $\frac{25}{2\times 257}$
• C. $\frac{35}{2\times 257}$
• D. $\frac{51}{2\times 257}$

Answer 1

Answer: (D)

$\frac{51}{2\times 257}$

$T_n=\frac{1}{(2+(n-1\left)5\right)(7+(n-1\left)5\right)}=\frac{1}{(5n-3)(5n+2)}{T}_n=\frac{1}{5}[\frac{1}{(5n-3)}-\frac{1}{(5n+2)}]T_1=\frac{1}{5}[\frac{1}{2}-\frac{1}{7}]T_2=\frac{1}{5}[\frac{1}{7}-\frac{1}{12}]T_3=\frac{1}{5}[\frac{1}{12}-\frac{1}{17}]\underset{–}{T_n=\frac{1}{5}[\frac{1}{(5n-3)}-\frac{1}{(5n+2)}]}{S}_n=\frac{1}{5}[\frac{1}{2}-\frac{1}{(5n+2)}]=\frac{5n+2-2}{10(5n+2)}{S}_n=\frac{n}{2(5n+2)}5n+2=257\Rightarrow n=51S_n=\frac{51}{2\times 257}$