Let-n-be-positive-integer-satisfies-a-n-1-1-n-1-n-1-1-n-1-n-1-Find-the-value-of-a-1-a-2-a-3-a-99-

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

Let n be positive integer satisfies

a_{n} = 1 + \sqrt{\frac{1}{n}} - \sqrt{\frac{1}{n + 1}} - \sqrt{\frac{1}{n} - \frac{1}{n + 1}}

Find the value of

a_{1} a_{2} a_{3} \dots a_{99}

Commented by Frix last updated on 24/Jul/24

I found this :

\underset{k = 1}{\prod^{n (n + 2)}} (a_{k}) = \frac{2}{(n + 2) (n + 1)}

99 = n (n + 2) \Rightarrow n = 9 \Rightarrow answer is \frac{1}{55}

Commented by naka3546 last updated on 24/Jul/24

Thanks