\frac1{1+\sqrt2}+\frac1{\sqrt2+\sqrt3}+\frac1{\sqrt3{+}{\sqrt4}}+\frac1{\sqrt4{+}{\sqrt5}}+…+\frac1{\sqrt{2020}+\sqrt{2021}}=4041
ශ්රේණියේ ඓක්ය එකවර ලියන්න ඔයාටත් පුළුවන්ද?
ප්රතිබද්ධයෙන් ගුණ කිරීමෙන් ,\begin{array}{rcl}\frac{\displaystyle1}{\displaystyle\;\sqrt{\mathrm k}+\;\sqrt{\mathrm k+1}}\;&=&\frac{\displaystyle1}{\displaystyle\;\sqrt{\mathrm k}+\;\sqrt{\mathrm k+1}}\;\times\frac{\sqrt{\mathrm k+1}{\displaystyle\;}{\displaystyle-}{\displaystyle\;}{\displaystyle\sqrt{\mathrm k}}}{\sqrt{\mathrm k+1}{\displaystyle\;}{\displaystyle-}{\displaystyle\;}{\displaystyle\sqrt{\mathrm k}}}\\\;\;\;\;\;\;\;&=&\frac{\sqrt{\mathrm k+1}{\displaystyle\;}{\displaystyle-}{\displaystyle\;}{\displaystyle\sqrt{\mathrm k}}}{(\mathrm k\;+\;1)\;-\;\mathrm k}\\{\mathrm U}_{\mathrm k}\;\;&=&\;-\sqrt{\mathrm k}\;+\;\sqrt{\mathrm k+1}\\\frac{\displaystyle1}{\displaystyle1+\;\sqrt2}\;+\;\frac{\displaystyle1}{\displaystyle\;\sqrt2+\;\sqrt3}\;+\frac{\displaystyle1}{\displaystyle\;\sqrt3+\;\sqrt4}\;+\;\frac{\displaystyle1}{\displaystyle\;\sqrt4+\;\sqrt5}\;+\;...................+\;\frac{\displaystyle1}{\displaystyle\;\sqrt{2020}+\;\sqrt{2021}}\;\;&=&\;\left(-\sqrt1\;+\;\sqrt2\right)\;+\;\left(-\sqrt2\;+\;\sqrt3\right)\;+\;\left(-\sqrt3\;+\;\sqrt4\right)\;+........+\left(-\sqrt{2020}\;+\;\sqrt{2021}\right)\\&=&\;\underline{-1\;+\;\sqrt{2021}}\\&&\end{array}
