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Question 1166028: Calculate
(1/3 × 7) + (1/7 × 11) + (1/11 × 15)+ · · · + (1/95 × 99).
Answer by ikleyn(52905)   (Show Source):
You can put this solution on YOUR website! .
Unfortunately, you incorrectly wrote (presented) the problem in your post.
The correct formulation is THIS:
calculate 1/(3 × 7) + 1/(7 × 11) + 1/(11 × 15) + · · · + 1/(95 × 99).
Do not ask me on "how I know it ?". Simply take it as a fact of the nature.
Solution
Write the separate terms of the series one after another
 = 
 = 
 = 
. . . and so on . . .
 = 
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  . ANSWER
The final formula can be simplified. I leave it to you to complete it, if it is required.
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: 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, when I shared my knowledge with you, you should be able to understand it.
For it, try to strain your mind, 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 ONLY ONE : to turn on and to strain your mind.
After I shared my knowledge with you, it is YOUR TURN now to make a step.
If you are not able to do it, it means that you are in wrong class and this problem is not for you.
Have a nice day (!)
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