SOLUTION: find the partial fraction decomposition for 2/(x(x+2)) and use the result to find the following sum 2/(1*3) + 2/(3*5) + 2/(5*7) + ... + 2/(99*101)

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: find the partial fraction decomposition for 2/(x(x+2)) and use the result to find the following sum 2/(1*3) + 2/(3*5) + 2/(5*7) + ... + 2/(99*101)      Log On


   

Question 1117401: find the partial fraction decomposition for 2/(x(x+2)) and use the result to find the following sum
2/(1*3) + 2/(3*5) + 2/(5*7) + ... + 2/(99*101)
Answer by ikleyn(52914) About Me  (Show Source):
You can put this solution on YOUR website!
.
2%2F%28x%2A%28x%2B2%29%29 = 1%2Fx - 1%2F%28x%2B2%29.


So, 


  2%2F%281%2A3%29  + 2%2F%283%2A5%29 + 2%2F%285%2A7%29  + ...   + 2%2F%2899%2A101%29 = 


= %281%2F1+-+1%2F3%29 + 1%2F3+-+1%2F5%29 + %281%2F5-1%2F7%29 + . . . + %281%2F99-1%2F101%29 = 


= all intermediate terms cancel each other, which leaves = 1+-+1%2F101 = 100%2F101.

Solved.

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These tricks are very well known.  See the lesson
    - Calculations with fractions
in this site.