Mathematics
AundreaLafaver57
13

Compute the sum \[\frac{1}{\sqrt{100} + \sqrt{102}} + \frac{1}{\sqrt{102} + \sqrt{104}} + \frac{1}{\sqrt{104}+\sqrt{106}} + \cdots + \frac{1}{\sqrt{9998} + \sqrt{10000}}.\]

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(1) Answers
jonathan1

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

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