Question

Ma puteti ajuta cu acest exercitiu? Va rog frumos!

Answer 1

$A = \dfrac{1}{\sqrt1+\sqrt2} + \dfrac{1}{\sqrt2+\sqrt3}+...+\dfrac{1}{\sqrt{99}+\sqrt{100}} \\ \text{Rationalizam si o sa avem} \\ \\ A = \dfrac{\sqrt1-\sqrt2}{1-2} + \dfrac{\sqrt2-\sqrt3}{2-3}+...+\dfrac{\sqrt{99}-\sqrt{100}}{99-100}}\\ \\ A = \dfrac{\sqrt1-\sqrt2}{-1} + \dfrac{\sqrt2-\sqrt3}{-1}+...+\dfrac{\sqrt{99}-\sqrt{100}}{-1}} \\ \\ \\ A = -\sqrt1+\sqrt2 -\sqrt2+\sqrt3-....-\sqrt{99}+\sqrt{100} \\ \\ \text{O sa se reduca teremnii si o sa ramana } \to \sqrt1~\text{si}~\sqrt{100}$$\downarrow \\ \\ \boxed{A = -\sqrt1+\sqrt{100}} \\ \\ \boxed{A = -1 + 10} \\ \\ \boxed{A = 9 \in \mathbb{N}}$

Answer 2

$A = \sqrt{2} - \sqrt{1} + \sqrt{3} - \sqrt{2} + ... + \sqrt{100} - \sqrt{99} \\ \\ A = \sqrt{100} - \sqrt{1} \\ \\ A = 10 - 1$

A=9 € N