2. Zadatak

Racionalizirajmo nazivnik u svakom članu danog zbroja. Imamo da je

$$\begin{aligned} \frac1{\sqrt1+\sqrt2}&=\frac1{\sqrt1+\sqrt2}\cdot\frac{\sqrt1-\sqrt2}{\sqrt1-\sqrt2}=\frac{\sqrt1-\sqrt2}{1-2}\\ &=\frac{\sqrt1-\sqrt2}{-1}=-\sqrt1+\sqrt2\\ \frac1{\sqrt2+\sqrt3}&=\frac1{\sqrt2+\sqrt3}\cdot\frac{\sqrt2-\sqrt3}{\sqrt2-\sqrt3}=\frac{\sqrt2-\sqrt3}{2-3}\\ &=\frac{\sqrt2-\sqrt3}{-1}=-\sqrt2+\sqrt3\\ \frac1{\sqrt3+\sqrt4}&=\frac1{\sqrt3+\sqrt4}\cdot\frac{\sqrt3-\sqrt4}{\sqrt3-\sqrt4}=\frac{\sqrt3-\sqrt4}{3-4}\\ &=\frac{\sqrt3-\sqrt4}{-1}=-\sqrt3+\sqrt4\\ &\cdots\\ \frac1{\sqrt{99}+\sqrt{100}}&=\frac1{\sqrt{99}+\sqrt{100}}\cdot\frac{\sqrt{99}-\sqrt{100}}{\sqrt{99}-\sqrt{100}}\\ &=\frac{\sqrt{99}-\sqrt{100}}{99-100}=\frac{\sqrt{99}-\sqrt{100}}{-1}\\ &=-\sqrt{99}+\sqrt{100} \end{aligned}$$

Prema tome, traženi zbroj je jednak

$$\begin{aligned} &-\sqrt1+\sqrt2-\sqrt2+\sqrt3-\sqrt3+\sqrt4-\cdots-\sqrt{99}+\sqrt{100}\\ &=-\sqrt1+\sqrt{100}=-1+10=9, \end{aligned}$$

budući se u danom izrazu krate svi članovi osim prvog i posljednjeg.