Question

Calculati $\frac{1}{ \sqrt{5}+ \sqrt{3} }+ \frac{1}{ \sqrt{7}+ \sqrt{5} } + ... \frac{1}{ \sqrt{99} + \sqrt{97}}$
Va rog mult, am nevoie .

Answer 1

$\text{Cel mai simplu este sa rationalizam toti numitorii.}\\ \frac{\sqrt5-\sqrt 3}{(\sqrt5+\sqrt3)(\sqrt5-\sqrt3)}+\frac{\sqrt 7-\sqrt 5}{(\sqrt 7+\sqrt 5)(\sqrt 7-\sqrt 5)}+...+\frac{\sqrt {99}-\sqrt{97} }{(\sqrt {99}+\sqrt{97})(\sqrt{99}-\sqrt{97})}=\\ \\ \dfrac{\sqrt5-\sqrt3}{5-3}+\dfrac{\sqrt 7-\sqrt 5}{7-5}+\ldots+\dfrac{\sqrt{99}-\sqrt{97}}{99-97}=\\ \dfrac{\sqrt 5-\sqrt 3}{2}+\dfrac{\sqrt 7-\sqrt 5}{2}+\ldots+\dfrac{\sqrt{99}-\sqrt{97}}{2}=$

$\dfrac{1}{2}\cdot \left(\sqrt 5-\sqrt 3+\sqrt7 -\sqrt 5+\ldots +\sqrt{99}-\sqrt{97}\right)=\boxed{\dfrac{\sqrt{99}-\sqrt 3}{2}}$