Question Number 123661 by weltr last updated on 27/Nov/20
$${find}\:{sum}\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\sqrt{\mathrm{2}}}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}\:+\:\sqrt{\mathrm{3}}}\:+\:...\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2019}}\:+\:\sqrt{\mathrm{2020}}} \\ $$
Answered by Dwaipayan Shikari last updated on 27/Nov/20
$$\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}}}=\sqrt{\mathrm{2}}−\mathrm{1} \\ $$$$\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}+\sqrt{\mathrm{3}}}=\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}} \\ $$$${sum}\:\:=\:\sqrt{\mathrm{2}}−\mathrm{1}+\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}+\sqrt{\mathrm{4}}−\sqrt{\mathrm{3}}+...+\sqrt{\mathrm{2020}}−\sqrt{\mathrm{2019}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:\sqrt{\mathrm{2020}}−\mathrm{1} \\ $$
Commented by weltr last updated on 27/Nov/20
$${thanks},\:{but}\:{where}\:{did}\:{you}\:{get}\:\sqrt{\mathrm{4}}\:−\sqrt{\mathrm{3}}\:{from}? \\ $$
Commented by Dwaipayan Shikari last updated on 27/Nov/20
$${Other}\:{terms}\:{are}\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{4}}+\sqrt{\mathrm{3}}},\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{4}}+\sqrt{\mathrm{5}}}...{which}\:{are}\:\sqrt{\mathrm{4}}−\sqrt{\mathrm{3}},\:\sqrt{\mathrm{5}}−\sqrt{\mathrm{4}}..... \\ $$