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For a positive integer  let

\(\[a_n = 1 + \sqrt{\frac{1}{n}} - \sqrt{\frac{1}{n + 1}} - \sqrt{\frac{1}{n} - \frac{1}{n + 1}}.\]\)
Compute the product 

(A)1/55  (B)1/110  (C)1/99  (D)2/99  (E) 1/100

 Sep 4, 2023
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We can simplify the expression for an​ as follows:

a_n = 1 + sqrt(1/n) - sqrt(1/(n + 1)) - sqrt(1/n - 1/(n + 1)) = 1 + sqrt(1/n) - sqrt(1/n + 1/(n(n + 1))) - sqrt(1/n - 1/(n(n + 1))) = 1 + sqrt(1/n) - sqrt((n + 1)/(n(n + 1))) - sqrt((n - 1)/(n(n + 1))) = 1 + sqrt(1/n) - (n + 1)/n - (n - 1)/(n + 1) = (n + 1)/(n + 1) - (n + 1)/n - (n - 1)/(n + 1) = 1 - 1/n

 

Then the product a_1 a_2 a_3 works out to 1/99.  The answer is (C).

 Sep 4, 2023

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