SOLUTION: Compute the sum {1}/{sqrt{100} + sqrt{102}} + {1}/{sqrt{102} + sqrt{104}} + {1}/{sqrt{104}+sqrt{106}} + ...+ {1}/{sqrt{9998} + sqrt{10000}}.

Algebra ->  Permutations -> SOLUTION: Compute the sum {1}/{sqrt{100} + sqrt{102}} + {1}/{sqrt{102} + sqrt{104}} + {1}/{sqrt{104}+sqrt{106}} + ...+ {1}/{sqrt{9998} + sqrt{10000}}.       Log On


   

Question 1066672: Compute the sum
{1}/{sqrt{100} + sqrt{102}} + {1}/{sqrt{102} + sqrt{104}} + {1}/{sqrt{104}+sqrt{106}} + ...+ {1}/{sqrt{9998} + sqrt{10000}}.
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
1%2F%28sqrt%28100%29+%2B+sqrt%28102%29%29 = multiply the numerator and denominator by %28sqrt%28100%29+-+sqrt%28102%29%29 (rationalizing the denominator) to get = %28sqrt%28100%29+-+sqrt%28102%29%29%2F%28-2%29.


Do it with every atom in your long molecule.

Then add.


Everything inside will cancel.


Only two terms survive: very first and very last.


So, the sum is


sqrt%28100%29%2F%28-2%29 - sqrt%2810000%29%2F%28-2%29 = 10%2F%28-2%29+-+100%2F%28-2%29 = -5 - (-50) = 45.

SOLVED.