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Question Number 209881 by naka3546 last updated on 24/Jul/24

$$\mathrm{Let}n\mathrm{be}\mathrm{positive}\mathrm{integer}\mathrm{satisfies}$$$$$$$$a_n=1+\sqrt{\frac{1}{n}}-\sqrt{\frac{1}{n+1}}-\sqrt{\frac{1}{n}-\frac{1}{n+1}}$$$$$$$$\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}$$$$$$$$a_1{a}_2{a}_3\dots a_{99}$$

Commented by Frix last updated on 24/Jul/24

$$\mathrm{I}\mathrm{found}\mathrm{this}:$$$$\underset{k=1}{\prod ^{n(n+2)}}\left(a_k\right)=\frac{2}{(n+2)(n+1)}$$$$99=n(n+2)\Rightarrow n=9\Rightarrow \mathrm{answer}\mathrm{is}\frac{1}{55}$$

Commented by naka3546 last updated on 24/Jul/24

$$\mathrm{Thanks}$$

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