Question

С подробным решением!

Answer 1

$\frac{1}{\sqrt{2}+1 }+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+...+\frac{1}{\sqrt{2025}+\sqrt{2024}}= \\\\=\frac{\sqrt{2}-1 }{(\sqrt{2}+1 )(\sqrt{2}-1) }+\frac{\sqrt{3} -\sqrt{2} }{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}+\frac{\sqrt{4}-\sqrt{3}}{(\sqrt{4}+\sqrt{3})(\sqrt{4} -\sqrt{3}) }+...+\frac{\sqrt{2025}-\sqrt{2024}}{(\sqrt{2025}+\sqrt{2024})(\sqrt{2025-\sqrt{2024})}}=\\\\=\sqrt{2} -1+\sqrt{3}-\sqrt{2}+\sqrt{4} -\sqrt{3} +...+\sqrt{2025}-\sqrt{2024} =-1+\sqrt{2025}=-1+45=$

$=\boxed{44}$