Q57
ArithmeticAlgebra, Equations & Operations
The value of $\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} \cdots + \frac{1}{\sqrt{80} + \sqrt{81}}$ is
- a.9
- b.7
- c.8
- d.$\sqrt{80}$
Answer: (C) 8
Rationalising each term gives $\frac{1}{\sqrt{n} + \sqrt{n + 1}} = \sqrt{n + 1} - \sqrt{n}$, so the sum telescopes. Everything cancels except $\sqrt{81} - \sqrt{1} = 9 - 1 = 8$. Option A forgets to subtract $\sqrt{1}$.