\(\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+...+\dfrac{1}{\sqrt{899}+\sqrt{900}}\)
\(=\dfrac{\sqrt{2}-\sqrt{1}}{\left(\sqrt{2}+\sqrt{1}\right)\left(\sqrt{2}-\sqrt{1}\right)}+\dfrac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}+...+\dfrac{\sqrt{900}-\sqrt{899}}{\left(\sqrt{900}+\sqrt{899}\right)\left(\sqrt{900}-\sqrt{899}\right)}\)
\(=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+...+\sqrt{900}-\sqrt{899}\)
\(=\sqrt{900}-1=29\)
Xét về tính tổng quát:
\(\dfrac{1}{\sqrt{k}+\sqrt{k+1}}=\dfrac{\sqrt{k+1}-\sqrt{k}}{\left(\sqrt{k+1}-\sqrt{k}\right)\left(\sqrt{k+1}+\sqrt{k}\right)}=\dfrac{\sqrt{k+1}-\sqrt{k}}{k+1-k}\)
\(=\sqrt{k+1}-\sqrt{k}\)
\(=>\dfrac{1}{\sqrt{1}+\sqrt{2}}=\sqrt{2}-1\)
=> Biểu thức trở thành:
\(\sqrt{2}-1+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}...+\sqrt{900}-\sqrt{899}\)
\(=\sqrt{900}-1=30-1=29\)
