Question

Обчислити $frac{1}{ sqrt{1} + sqrt{3} } + frac{1}{ sqrt{3} + sqrt{5}}+...+ frac{1}{ sqrt{119}+ sqrt{121} }$

Answer 1

$frac{1}{sqrt1+sqrt3} + frac{1}{sqrt3+sqrt5} +...+ frac{1}{sqrt{119}+sqrt{121}} =\\= frac{sqrt1-sqrt3}{(sqrt1+sqrt3)(sqrt1-sqrt3)} +frac{sqrt3-sqrt5}{(sqrt3+sqrt5)(sqrt3-sqrt5)}+...+frac{sqrt{119}-sqrt{121}}{(sqrt{119}+sqrt{121})(sqrt{119}-sqrt{121})}=\\=frac{sqrt1-sqrt3}{1-3}+ frac{sqrt3-sqrt5}{3-5}+...+ frac{sqrt{119}-sqrt{121}}{119-121}=\\= frac{sqrt1-sqrt3}{-2} +frac{sqrt3-sqrt5}{-2} +...+ frac{sqrt{119}-sqrt{121}}{-2}=$

$=- frac{1}{2}cdot (sqrt1-sqrt3+sqrt3-sqrt5+sqrt5-sqrt7+...+sqrt{117}-sqrt{119}+\\+sqrt{119}-sqrt{121})= - frac{1}{2}cdot (sqrt1-sqrt{121})=- frac{1}{2}cdot (1-11)=frac{10}{2}=5$