\(A=\dfrac{\sqrt{4}-1}{\left(\sqrt{4}-1\right)\left(\sqrt{4}+1\right)}+\dfrac{\sqrt{7}-\sqrt{4}}{\left(\sqrt{7}-\sqrt{4}\right)\left(\sqrt{7+\sqrt{4}}\right)}+...+\dfrac{\sqrt{100}-\sqrt{97}}{\left(\sqrt{100}-\sqrt{97}\right)\left(\sqrt{100}+\sqrt{97}\right)}\)
\(A=\dfrac{\sqrt{4}-\sqrt{1}}{3}+\dfrac{\sqrt{7}-\sqrt{4}}{3}+...+\dfrac{\sqrt{100}-\sqrt{97}}{3}\)
\(A=\dfrac{1}{3}\left(\sqrt{4}-\sqrt{1}+\sqrt{7}-\sqrt{4}+...+\sqrt{100}-\sqrt{97}\right)\)
\(A=\dfrac{1}{3}\left(\sqrt{100}-\sqrt{1}\right)=\dfrac{9}{3}=3\)
Lời giải:
\(3A=\frac{3}{1+\sqrt{4}}+\frac{3}{\sqrt{4}+\sqrt{7}}+\frac{3}{\sqrt{7}+\sqrt{10}}+...+\frac{1}{\sqrt{97}+\sqrt{100}}\)
\(=\frac{(\sqrt{4}-1)(\sqrt{4}+1)}{1+\sqrt{4}}+\frac{(\sqrt{7}-\sqrt{4})(\sqrt{7}+\sqrt{4})}{\sqrt{4}+\sqrt{7}}+\frac{(\sqrt{10}-\sqrt{7})(\sqrt{10}+\sqrt{7})}{\sqrt{7}+\sqrt{10}}+.....+\frac{(\sqrt{100}-\sqrt{97})(\sqrt{100}+\sqrt{97})}{\sqrt{97}+\sqrt{100}}\)
\(=(\sqrt{4}-1)+(\sqrt{7}-\sqrt{4})+(\sqrt{10}-\sqrt{7})+...+(\sqrt{100}-\sqrt{97})\)
\(=\sqrt{100}-1=10-1=9\)
\(\Rightarrow A=3\)
