What is the value of the following? \frac{1}{5\sqrt{4}+4\sqrt{5}}+\frac{1}{6\sqrt{5}+5\sqrt{6}}+\frac{1}{7\sqrt{6}+6\sqrt{7}}+\frac{1}{8\sqrt{7}+7\sqrt{8}}+\frac{1}{9\sqrt{8}+8\sqrt{9}}

  1. A. \frac{1}{\sqrt{6}}
  2. B. \frac{1}{2}
  3. C. 1
  4. D. \frac{1}{6} ✓

Correct Answer: D. \frac{1}{6}

Explanation

The general term \frac{1}{(n+1)\sqrt{n}+n\sqrt{n+1}} rationalizes to \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}}. This forms a telescoping series from n=4 to n=8. The sum is \frac{1}{\sqrt{4}} - \frac{1}{\sqrt{9}} = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}.

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