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
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).
