Evaluate $$\frac{\sum_{n=1}^{99}\sqrt{10+\sqrt{n}}}{\sum_{n=1}^{99}\sqrt{10-\sqrt{n}}}$$
Both the top and bottom do not telescope and and last and first do not add up to something nice, so I do not know how to proceed. Thanks!
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If $k$ is the result of the expression, then we have $\sum_{n=1}^{99}\sqrt{10+\sqrt n}-k\sqrt{10-\sqrt n}=0$, but this doesn't do much to improve the situation... – abiessu Dec 03 '19 at 21:47
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check https://math.stackexchange.com/questions/3404396/simplify-frac-sqrt10-sqrt1-sqrt10-sqrt2-ldots-sqrt10-sqrt99/3404433#3404433 – Quanto Dec 03 '19 at 22:03