[tex] \frac{1}{ \sqrt{100}+ \sqrt{102} } + \frac{1}{ \sqrt{102} + \sqrt{104} }+...+ \frac{1}{ \sqrt{9998}+ \sqrt{10000} } [/tex]
We will multiply all fractions to make a difference of squares in the denominators. So this sum will become:
[tex] \frac{ \sqrt{102} - \sqrt{100} }{2}+ \frac{ \sqrt{104}- \sqrt{102} }{2}+...+ \frac{ \sqrt{10000}- \sqrt{9998} }{2} [/tex]
= - √100 / 2 + √10000 / 2 = - 10 / 2 + 100 / 2 =
= - 5 + 50 = 45
