Question

Calcular a soma S = $\frac{1}{ \sqrt{2}+ \sqrt{1} } + \frac{1}{ \sqrt{3}+ \sqrt{2} } + \frac{1}{ \sqrt{4}+ \sqrt{3} } +...+ \frac{1}{ \sqrt{81} + \sqrt{80} }$

Answer 1

$\frac{1}{ \sqrt{x+1}+ \sqrt{x} } =\sqrt{x+1}- \sqrt{x} \\ \\ \frac{1}{ \sqrt{2}+ \sqrt{1} } + \frac{1}{ \sqrt{3}+ \sqrt{2} } + \frac{1}{ \sqrt{4}+ \sqrt{3} } +...+ \frac{1}{ \sqrt{81} + \sqrt{80} } = \\ \sqrt{2} - \sqrt{1} + \sqrt{3} - \sqrt{2} + \sqrt{4} - \sqrt{3} +.....+ \sqrt{81} - \sqrt{80} = \\ =- \sqrt{1} + \sqrt{81} =-1+9=8$