Question

find the value of S=1/2*7+1/7*12+1/12*17+.....+1/252*257

Answer 1

Hope this will help you.
Sorry for bad handwriting, I was in a hurry.

Answer 2

The sum of this series would be 51/514

1)  The given series is 1/2*7 , 1/7*12, 1/12*17.....

2) General nth term of this series can be written as 1/ (5n-3)(5n+2)

3) This nth term can be written as :$\frac{1}{5}*(\frac{1}{5n-3} - \frac{1}{5n+2} )$

4) This can be resolved as: $\frac{1}{5} *\frac{1}{5n-3} - \frac{1}{5} *\frac{1}{5n+2}$

5)  Hence, first term would be 1/5( 1/2 -1/7)

Second term would be 1/5 (1/7 - 1/12)

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Last term would be 1/5 ( 1/252 -1/257)

6) Summing all these terms would give 1/5 (1/2-1/7 +1/7 -1/12 +1/12 -1/17 ........ -1/257)

= 1/5( 1/2 -1/257)      { All other terms will get canceled)

= 1/5( 255/514) = 51/514