Question 1166028
.


            Unfortunately,  you incorrectly wrote  (presented)  the problem in your post.


            The correct formulation is  THIS: 


<pre>
                calculate  1/(3 × 7) + 1/(7 × 11) + 1/(11 × 15) + · · · + 1/(95 × 99).
</pre>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Do not ask me on &nbsp;"how I know it ?".  &nbsp;Simply take it as a fact of the nature.



<U>Solution</U>


<pre>
Write the separate terms of the series one after another


{{{1/(3 * 7)}}} = {{{(1/4)*(1/3 - 1/7)}}}


{{{1/(7 * 11)}}} = {{{(1/4)*(1/7 - 1/11)}}}


{{{1/(11 * 15)}}} = {{{(1/4)*(1/11 - 1/15)}}}


   . . . and so on . . . 


{{{1/(95 * 99)}}} = {{{(1/4)*(1/95 - 1/99)}}}



Now sum up all these lines.


On the left, you will get the seeking sum.


On the right, all interior terms/pieces with opposite signs will cancel each other, leaving survived 
only very first and very last terms/pieces.


Therefore, the seeking sum is equal to  {{{(1/4)*(1/3 - 1/99)}}}.    <U>ANSWER</U>


The final formula can be simplified.  I leave it to you to complete it, if it is required.
</pre>

The solution is completed.



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Comment from student: How do you know that all interior terms will cancel each other?




My response: &nbsp;&nbsp;there are two causes for it.


First cause is that I am an expert.


Second cause is the it is an obvious / (self-evident) fact.



Now, &nbsp;when I shared my knowledge with you, &nbsp;you should be able to understand it.


For it, &nbsp;try to strain your mind, &nbsp;looking into the formulas.



Let me tell you one thing.



This problem is half-level upper the average school assignments.

It is intendent for those students who like to strain their mind to learn something new.

The way for it is &nbsp;ONLY &nbsp;ONE : &nbsp;to turn on and to strain your mind.


After I shared my knowledge with you, &nbsp;it is &nbsp;YOUR &nbsp;TURN &nbsp;now to make a step.


If you are not able to do it, &nbsp;it means that you are in wrong class and this problem is not for you.



Have a nice day &nbsp;(!)