If frac{1}{2 times 7} + frac{1}{7 times 12} + | vedclass

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(C) The given series is $S = \sum_{n=1}^{10} \frac{1}{a_n 	imes b_n}$,where $a_n = 5n - 3$ and $b_n = 5n + 2$.We can write the general term as $T_n =

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$\frac{5}{52}$

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(C) The given series is $S = \sum_{n=1}^{10} \frac{1}{a_n 	imes b_n}$,where $a_n = 5n - 3$ and $b_n = 5n + 2$.We can write the general term as $T_n = \frac{1}{(5n-3)(5n+2)}$.Using partial fractions,$T_n = \frac{1}{5} \left( \frac{1}{5n-3} - \frac{1}{5n+2} ight)$.Summing from $n=1$ to $10$:$S = \frac{1}{5} \left[ (\frac{1}{2} - \frac{1}{7}) + (\frac{1}{7} - \frac{1}{12}) + \dots + (\frac{1}{47} - \frac{1}{52}) ight]$.This is a telescoping series,so $S = \frac{1}{5} \left( \frac{1}{2} - \frac{1}{52} ight)$.$S = \frac{1}{5} \left( \frac{26-1}{52} ight) = \frac{1}{5} 	imes \frac{25}{52} = \frac{5}{52}$.

If $\frac{1}{2 \times 7} + \frac{1}{7 \times 12} + \frac{1}{12 \times 17} + \frac{1}{17 \times 22} + \dots$ to $10$ terms $= k$,then $k =$

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