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Simplify $$\frac{\sqrt{45 + \sqrt{1}} + \sqrt{45 + \sqrt{2}} + \sqrt{45 + \sqrt{3}} + \dots + \sqrt{45 + \sqrt{2024}}}{\sqrt{45 - \sqrt{1}} + \sqrt{45 - \sqrt{2}} + \sqrt{45 - \sqrt{3}} + \dots + \sqrt{45 - \sqrt{2024}}}.$$

Feb 25, 2019

#1
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sumfor(n, 1, 2024, sqrt(45 + sqrt(n)) / sqrt(45 - sqrt(n))=7098.9283254388............

Feb 26, 2019
#2
+24983
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Simplify

$$\mathbf{x=\dfrac{\sum \limits_{n = 1}^{2024} \sqrt{45 + \sqrt{n}} }{\sum \limits_{n = 1}^{2024} \sqrt{45 - \sqrt{n}} } = \dfrac{\sqrt{45 + \sqrt{1}} + \sqrt{45 + \sqrt{2}} + \sqrt{45 + \sqrt{3}} + \dots + \sqrt{45 + \sqrt{2024}}}{\sqrt{45 - \sqrt{1}} + \sqrt{45 - \sqrt{2}} + \sqrt{45 - \sqrt{3}} + \dots + \sqrt{45 - \sqrt{2024}}}}$$.

1.

$$\begin{array}{|rcll|} \hline \Big( \sqrt{45 + \sqrt{n}} + \sqrt{45 - \sqrt{n}} \Big)^2 &=& 45 + \sqrt{n} +2\sqrt{45 + \sqrt{n}}\sqrt{45 - \sqrt{n}} +45 -\sqrt{n} \\ &=& 90 +2\sqrt{(45 + \sqrt{n} )(45 - \sqrt{n}) } \\ &=& 90 +2\sqrt{45^2-n } \\ &=& 2\cdot \Big( 45 + \sqrt{2025-n }\Big) \\\\ \sqrt{45 + \sqrt{n}} + \sqrt{45 - \sqrt{n}} &=& \sqrt{2}\sqrt{ 45 + \sqrt{2025-n} } \\ \mathbf{\sqrt{ 45 + \sqrt{2025-n} } } &\mathbf{=} & \mathbf{\dfrac{1}{\sqrt{2}}\Big( \sqrt{45 + \sqrt{n}} + \sqrt{45 - \sqrt{n}} \Big) }~ \huge{\text{!}} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{x=\dfrac{\sum \limits_{n = 1}^{2024} \sqrt{45 + \sqrt{n}} }{\sum \limits_{n = 1}^{2024} \sqrt{45 - \sqrt{n}} } } &=& \mathbf{\dfrac{\sum \limits_{n = 1}^{2024} \sqrt{45 + \sqrt{2025-n}} }{\sum \limits_{n = 1}^{2024} \sqrt{45 - \sqrt{n}} } } \\\\ &=& \dfrac {\dfrac{1}{\sqrt{2}}\sum \limits_{n = 1}^{2024} \Big( \sqrt{45 + \sqrt{n}} + \sqrt{45 - \sqrt{n}} \Big)} {\sum \limits_{n = 1}^{2024} \sqrt{45 - \sqrt{n}} } \\\\ &=& \dfrac{1}{\sqrt{2}} \cdot \dfrac {\sum \limits_{n = 1}^{2024} \Big( \sqrt{45 + \sqrt{n}} \Big)+\sum \limits_{n = 1}^{2024} \Big( \sqrt{45 - \sqrt{n}} \Big)} {\sum \limits_{n = 1}^{2024} \sqrt{45 - \sqrt{n}} } \\\\ &=& \dfrac{1}{\sqrt{2}} \cdot \left( 1+ \underbrace{ \dfrac {\sum \limits_{n = 1}^{2024} \Big( \sqrt{45 + \sqrt{n}} \Big)} {\sum \limits_{n = 1}^{2024} \sqrt{45 - \sqrt{n}} } }_{\huge{=x}} \right) \\\\ x &=& \dfrac{1}{\sqrt{2}} \cdot (1+x) \\\\ x\sqrt{2} &=& 1+x \\ x\sqrt{2} -x &=& 1 \\ x(\sqrt{2}-1) &=& 1 \\ x &=& \dfrac{1}{\sqrt{2}-1}\cdot \left(\dfrac{\sqrt{2}+1}{\sqrt{2}+1}\right) \\ x &=& \dfrac{\sqrt{2}+1}{2-1} \\ \mathbf{x} & \mathbf{=} & \mathbf{ \sqrt{2}+1 } \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \dfrac{\sqrt{45 + \sqrt{1}} + \sqrt{45 + \sqrt{2}} + \sqrt{45 + \sqrt{3}} + \dots + \sqrt{45 + \sqrt{2024}}}{\sqrt{45 - \sqrt{1}} + \sqrt{45 - \sqrt{2}} + \sqrt{45 - \sqrt{3}} + \dots + \sqrt{45 - \sqrt{2024}}} &=& \mathbf{ \sqrt{2}+1 } \\ \hline \end{array}$$

Feb 27, 2019
edited by heureka  Feb 27, 2019