Question

$\gray{ \underbrace{ \huge \underline{ \pink{ \sf \: Question:-}}}}$




$\begin{gathered} \sf \: if \: \sf\mathrm{x=\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}} \: \: and \: \: \mathrm{y=\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}} \\ \\ \\ \sf \: find \: the \: value \: of \: \sf{\dfrac{x^{2}+xy+y^{2}}{x^{2}-xy+y^{2}}}\end{gathered}$


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Answer 1

This is the first part rest is on the next answer
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Answer 2

Answer:

$begin{gathered}\sf{\implies \frac{1}{\sqrt n + \sqrt{n+1}} +\frac{1}{\sqrt{n+1} + \sqrt{n+2}} + ...}\\\\\sf{\implies\big( \frac{1}{\sqrt n + \sqrt{n+1}} \times \frac{\sqrt{n}-\sqrt{n+1}}{\sqrt{n} -\sqrt{n+1}}\big)+\big( \frac{1}{\sqrt{n+1} + \sqrt{n+2}}\times\frac{\sqrt{n+1}-\sqrt{n+2}}{\sqrt{n+1}-\sqrt{n+2}}\big) + ...}\\\\\sf{\implies \frac{\sqrt{n}-\sqrt{n+1}}{n-(n-1)}+\frac{\sqrt{n+1}-\sqrt{n+2}}{(n+1)-(n+2)} +... }}\\\\\sf{\implies(\sqrt{n+1}-\sqrt{n})+(\sqrt{n+2}-\sqrt{n+1})... }\end{gathered}$

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