Question

Tính :

a ) \(S=\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+.....+\dfrac{1}{\sqrt{100}+\sqrt{101}}\)

b ) \(S=\dfrac{1}{\sqrt{2}+\sqrt{4}}+\dfrac{1}{\sqrt{4}+\sqrt{6}}+...+\dfrac{1}{\sqrt{100}+\sqrt{102}}\)

Answer 1

a)

\(S=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+....+\frac{1}{\sqrt{100}+\sqrt{101}}\)

\(S=\frac{\sqrt{2}-\sqrt{1}}{(\sqrt{2}+\sqrt{1})(\sqrt{2}-\sqrt{1})}+\frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}+....+\frac{\sqrt{101}-\sqrt{100}}{(\sqrt{101}+\sqrt{100})(\sqrt{101}-\sqrt{100})}\)

\(S=\frac{\sqrt{2}-\sqrt{1}}{2-1}+\frac{\sqrt{3}-\sqrt{2}}{3-2}+...+\frac{\sqrt{101}-\sqrt{100}}{101-100}\)

\(S=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+...+\sqrt{101}-\sqrt{100}\)

\(S=\sqrt{101}-1\)

Answer 2

b)

\(S=\frac{1}{\sqrt{2}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{6}}+...+\frac{1}{\sqrt{100}+\sqrt{102}}\)

\(S=\frac{\sqrt{4}-\sqrt{2}}{(\sqrt{4}+\sqrt{2})(\sqrt{4}-\sqrt{2})}+\frac{\sqrt{6}-\sqrt{4}}{(\sqrt{6}+\sqrt{4})(\sqrt{6}-\sqrt{4})}+...+\frac{\sqrt{102}-\sqrt{100}}{(\sqrt{102}+\sqrt{100})(\sqrt{102}-\sqrt{100})}\)

\(S=\frac{\sqrt{4}-\sqrt{2}}{4-2}+\frac{\sqrt{6}-\sqrt{4}}{6-4}+....+\frac{\sqrt{102}-\sqrt{100}}{102-100}\)

\(S=\frac{\sqrt{4}-\sqrt{2}+\sqrt{6}-\sqrt{4}+\sqrt{8}-\sqrt{6}+...+\sqrt{102}-\sqrt{100}}{2}\)

\(S=\frac{\sqrt{102}-\sqrt{2}}{2}\)