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Question Number 203387 by Noorzai last updated on 18/Jan/24

Answered by mr W last updated on 18/Jan/24

$${(\sqrt{10+\sqrt{k}}-\sqrt{10-\sqrt{k}})}^2=2(10-\sqrt{100-k})$$$$\sqrt{10+\sqrt{k}}-\sqrt{10-\sqrt{k}}=\sqrt{2}\sqrt{10-\sqrt{100-k}}$$$$\underset{k=1}{\sum ^{99}}\sqrt{10+\sqrt{k}}-\underset{k=1}{\sum ^{99}}\sqrt{10-\sqrt{k}}=\sqrt{2}\underset{k=1}{\sum ^{99}}\sqrt{10-\sqrt{100-k}}$$$$\underset{k=1}{\sum ^{99}}\sqrt{10+\sqrt{k}}-\underset{k=1}{\sum ^{99}}\sqrt{10-\sqrt{k}}=\sqrt{2}\underset{k=1}{\sum ^{99}}\sqrt{10-\sqrt{k}}$$$$\underset{k=1}{\sum ^{99}}\sqrt{10+\sqrt{k}}=(\sqrt{2}+1)\underset{k=1}{\sum ^{99}}\sqrt{10-\sqrt{k}}$$$$\frac{\underset{k=1}{\sum ^{99}}\sqrt{10+\sqrt{k}}}{\underset{k=1}{\sum ^{99}}\sqrt{10-\sqrt{k}}}=\sqrt{2}+1✓$$

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