unifesp

por Mael0912 Sex 03 Nov 2023, 22:35

Unifesp (Adapt.) Calcule o valor da soma

por **Giovana Martins** Sex 03 Nov 2023, 23:02

Penso que seja isto!

[latex]\\\mathrm{\ \ \ \ \ \ \ \frac{1}{\sqrt{x}+\sqrt{y}}=\frac{\sqrt{x}-\sqrt{y}}{\left (\sqrt{x}+\sqrt{y} \right )\left ( \sqrt{x}-\sqrt{y} \right )}=\frac{\sqrt{x}-\sqrt{y}}{x-y}}\\\\ \mathrm{S=\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{999}+\sqrt{1000}}}\\\\ \mathrm{S=\frac{\sqrt{2}-1}{2-1}+\frac{\sqrt{3}-\sqrt{2}}{3-2}+\frac{\sqrt{4}-\sqrt{3}}{4-3}+...+\frac{\sqrt{1000}-\sqrt{999}}{1000-999}}\\\\ \mathrm{S=\cancel{\mathrm{\sqrt{2}}}-1+\cancel{\mathrm{\sqrt{3}}}-\cancel{\mathrm{\sqrt{2}}}+\cancel{\mathrm{\sqrt{4}}}-\cancel{\mathrm{\sqrt{3}}}+...+\sqrt{1000}-\cancel{\mathrm{\sqrt{999}}}}\\\\ \mathrm{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \boxed {\mathrm{S=\sqrt{1000}-1}}}[/latex]