Question

Plzz solve this question.

Answer 1

$\sf{Given : \displaystyle \sum\limits_{k = 4}^{143}\;\dfrac{1}{\sqrt{k} + \sqrt{k + 1}}}\\\\\\\textsf{Consider the Value of above Expression, When k = 4}\\\\\\\implies \dfrac{1}{\sqrt{4} + \sqrt{4 + 1}}\\\\\\\implies \dfrac{1}{\sqrt{4} + \sqrt{5}}\\\\\\\textsf{In order to Rationalize the Denominator :}\\ \\\textsf{We need to Multiply and Divide the above fraction with} : \sqrt{4} - \sqrt{5}}\\\\\\\implies \dfrac{\sqrt{4} - \sqrt{5}}{(\sqrt{4} + \sqrt{5})(\sqrt{4} - \sqrt{5})}$

$\sf{\implies \dfrac{\sqrt{4} - \sqrt{5}}{(\sqrt{4})^2 - (\sqrt{5})^2}$

$\sf{\implies \dfrac{\sqrt{4} - \sqrt{5}}{4 - 5}$

$\sf{\implies \dfrac{\sqrt{4} - \sqrt{5}}{-1}$

$\sf{\implies {-\sqrt{4} + \sqrt{5}}$

$\textsf{In the similar way : Consider the Value of above Expression, When k = 5}\\\\\\\sf{\implies \dfrac{1}{\sqrt{5} + \sqrt{5 + 1}}}\\\\\\\implies \dfrac{1}{\sqrt{5} + \sqrt{6}}\\\\\\\textsf{In order to Rationalize the Denominator :}\\ \\\textsf{We need to Multiply and Divide the above fraction with} : \sf{\sqrt{5} - \sqrt{6}}}\\\\\\\sf{\implies \dfrac{\sqrt{5} - \sqrt{6}}{(\sqrt{5} + \sqrt{6})(\sqrt{5} - \sqrt{6})}$

$\sf{\implies \dfrac{\sqrt{5} - \sqrt{6}}{(\sqrt{5})^2 - (\sqrt{6})^2}$

$\sf{\implies \dfrac{\sqrt{5} - \sqrt{6}}{5 - 6}$

$\sf{\implies \dfrac{\sqrt{5} - \sqrt{6}}{-1}$

$\sf{\implies {-\sqrt{5} + \sqrt{6}}$

$\textsf{From the above, We can notice that :}\\\\\bigstar\;\;\textsf{For\;k = 4, The Value of the given Expression is\;} \sf{-\sqrt{4} + \sqrt{5}}\\\\\bigstar\;\;\textsf{For\;k = 5, The Value of the given Expression is\;} \sf{-\sqrt{5} + \sqrt{6}}\\\\\bigstar\;\;\textsf{For\;k = 6, The Value of the given Expression is\;} \sf{-\sqrt{6} + \sqrt{7}}\\\\\textsf{In the similar way :}\\\\\bigstar\;\;\textsf{For\;k = 142, The Value of the given Expression is\;} \sf{-\sqrt{142} + \sqrt{143}}$

$\bigstar\;\;\textsf{For\;k = 143, The Value of the given Expression is\;} \sf{-\sqrt{143} + \sqrt{144}}$

$\sf{As\;\displaystyle \sum\limits_{k = 4}^{143}\;indicates : \;Sum\;of\;all\;values\;of\;given\;Expression\;at\;k= 4\;to\;k = 143}\\\\\\\sf{\implies The\;Value\;of\;\displaystyle \sum\limits_{k = 4}^{143}\;\dfrac{1}{\sqrt{k} + \sqrt{k + 1}}\;is :}\\\\\\\sf{\implies -\sqrt{4} + \sqrt{5} - \sqrt{5} + \sqrt{6}\;.\;.\;.-\sqrt{142} + \sqrt{143} -\sqrt{143} + \sqrt{144}}\\\\\\\sf{\implies -\sqrt{4} + \sqrt{144}}\\\\\\\sf{\implies - 2 + 12}\\\\\\\sf{\implies 10}$

$\sf{\implies 10 = a - \sqrt{b}}\\\\\\\textsf{Comparing both sides of the above Equation, We can notice that :}\\\\\\\sf{\implies a = 10\;and\;b = 0}$