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

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