SOLUTION: Find the sum of the series 24/2+24/6+24/12+24/20+24/30+...+24/4160

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Question 1147571: Find the sum of the series 24/2+24/6+24/12+24/20+24/30+...+24/4160
Answer by ikleyn(52866) About Me  (Show Source):
You can put this solution on YOUR website!
.

            In the shortest form, the solution is as follows.

            Take the factor 24 aside, for couple of minutes. 


The denominator to the n-th fraction is n*(n+1).


Therefore, your sequence is  (with the numerator replaced by 1, temporarily)


    1%2F%281%2A2%29 + 1%2F%282%2A3%29 + 1%2F3%2A4%29 + 1%2F%285%2A6%29 + . . . 1%2F%2864%2A65%29.


Each term is


    1%2F%281%2A2%29 = 1%2F1 - 1%2F2

    1%2F%282%2A3%29 = 1%2F2 - 1%2F3

    1%2F%283%2A4%29 = 1%2F3 - 1%2F4


     . . .  and so on  . . . 


     1%2F%28k%2A%28k%2B1%29%29 = 1%2Fk - 1%2F%28k%2B1%29

     . . .  and so on  . . . 

     1%2F%2864%2A65%29 = 1%2F64 - 1%2F65


Now add the fractions on the left and on the right sides.


You will get 


    the sum under the question = 24 multiplied by  ( 1%2F1 - 1%2F65 ),


since all other terms will cancel each other.



ANSWER.  24 - 24%2F65.


Solved.

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It is a typical school Math circle level problem.

To see many other similar solved problems, look into my lesson
    Calculations with fractions
in this site.