Question 1147571
.


            In the shortest form, the solution is as follows.


            Take the factor 24 aside, for couple of minutes.



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


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


    {{{1/(1*2)}}} + {{{1/(2*3)}}} + {{{1/3*4)}}} + {{{1/(5*6)}}} + . . . {{{1/(64*65)}}}.


Each term is


    {{{1/(1*2)}}} = {{{1/1}}} - {{{1/2}}}

    {{{1/(2*3)}}} = {{{1/2}}} - {{{1/3}}}

    {{{1/(3*4)}}} = {{{1/3}}} - {{{1/4}}}


     . . .  and so on  . . . 


     {{{1/(k*(k+1))}}} = {{{1/k}}} - {{{1/(k+1)}}}

     . . .  and so on  . . . 

     {{{1/(64*65)}}} = {{{1/64}}} - {{{1/65}}}


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


You will get 


    the sum under the question = 24 multiplied by  ( {{{1/1}}} - {{{1/65}}} ),


since all other terms will cancel each other.



<U>ANSWER</U>.  24 - {{{24/65}}}.
</pre>


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

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/NumericFractions/Calculations-with-fractions.lesson>Calculations with fractions</A>

in this site.