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If

$$u=\sqrt{c+1} - \sqrt{c}$$

and

$$v=\sqrt{c} - \sqrt{c-1}$$

then, which among $u$ and $v$ is greater?

Please help without substituting values.

  • 1
    Any thoughts? Hint: try a few values...that should at least tell you what the answer is. – 2017-02-24

3 Answers 3

2

$u=\frac{(\sqrt{c+1}-\sqrt{c})(\sqrt{c+1}+\sqrt{c})}{\sqrt{c}+\sqrt{c+1}} = \frac{1}{\sqrt{c}+\sqrt{c+1}}$

By analogy. $v=\frac{1}{\sqrt{c}+\sqrt{c-1}}$.

So,$v>u$

6

Hint: Multiply $u$ by $\frac {\sqrt{c+1}+\sqrt{c}}{\sqrt{c+1}+\sqrt{c}}$ and $v$ by $\frac {\sqrt{c-1}+\sqrt{c}}{\sqrt{c-1}+\sqrt{c}}$

6

Note that $\sqrt x$ is concave function, so $$2\sqrt x \gt \sqrt {x-1} + \sqrt {x+1}$$