if-a-gt-1-show-k-1-a-2-1-a-k-k-1-a-2-1-a-k-2-1-

Question Number 192009 by universe last updated on 05/May/23

i f a > 1, s h o w

\frac{\underset{k = 1}{\sum^{a^{2} - 1}} \sqrt{a + \sqrt{k}}}{\underset{k = 1}{\sum^{a^{2} - 1}} \sqrt{a - \sqrt{k}}} = \sqrt{2} + 1

Answered by Skabetix last updated on 05/May/23

Commented by Skabetix last updated on 05/May/23

S_{2} = S_{1} (\sqrt{} 2 - 1)

\to S_{1} = \frac{S_{2}}{\sqrt{2} - 1}

\to \frac{S_{1}}{S_{2}} = \frac{\frac{S_{2}}{\sqrt{2} - 1}}{\frac{S_{2}}{1}} = \frac{S_{2}}{\sqrt{2} - 1} \times \frac{1}{S_{2}} = \frac{1}{\sqrt{2} - 1} = \frac{\sqrt{2} + 1}{(\sqrt{} 2 - 1) (\sqrt{} 2 + 1)} = \sqrt{} 2 + 1

Answered by York12 last updated on 24/Jul/23

l e t \underset{k = 1}{\sum^{a^{2} - 1}} \sqrt{a + \sqrt{k}} = s_{1} a n d \underset{k = 1}{\sum^{a^{2} - 1}} \sqrt{a - \sqrt{k}} = s_{2}

s_{1} - s_{2} = \underset{k = 1}{\sum^{a^{2} - 1}} [\sqrt{{(\sqrt{a + \sqrt{k}} - \sqrt{a - \sqrt{k}})}^{2}}] = \underset{k = 1}{\sum^{a^{2} - 1}} \sqrt{2} \sqrt{a - \sqrt{a^{2} - k}} \to [I]

N o w s i n c e \underset{k = 1}{\sum^{a^{2} - 1}} T_{k} = \underset{k = 1}{\sum^{a^{2} - 1}} T_{[(a^{2} - 1) - (k - 1)]} = \underset{k = 1}{\sum^{a^{2} - 1}} T_{(a^{2} - k)}

l e t T_{k} = \sqrt{a - \sqrt{k}} \to T_{(a^{2} - k)} = \sqrt{a - \sqrt{a^{2} - k}}

∴ I = \underset{k = 1}{\sum^{a^{2} - 1}} \sqrt{a - \sqrt{k}} = \sqrt{2} s_{2}

∴ s_{1} + s_{2} = \sqrt{2} s_{2} \to s_{1} = (1 + \sqrt{2}) s_{2} \to \frac{s_{1}}{s_{2}} = (1 + \sqrt{2}) \to ({T h a t}^{'} s i t)

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