.
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)
+ + + + . . . .
Each term is
= -
= -
= -
. . . and so on . . .
= -
. . . and so on . . .
= -
Now add the fractions on the left and on the right sides.
You will get
the sum under the question = 24 multiplied by ( - ),
since all other terms will cancel each other.
ANSWER. 24 - .
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.