Will difference always works?

Calculus Level 3

$\begin{aligned} & \sqrt2 - \sqrt 1 \approx 0.4142\\& \sqrt3 - \sqrt 2 \approx 0.3178\\& \sqrt4 - \sqrt3\approx 0.2679 \\& \vdots\quad\vdots\quad\vdots\end{aligned}$

True or false

There exist positive integer $n$ such that $\sqrt{n+1}- \sqrt n > \dfrac{3}{5}$ .

false True

1 solution

Chew-Seong Cheong
Feb 17, 2018

$\begin{aligned} X & = \sqrt{n+1}-\sqrt n \\ & = \frac {(\sqrt{n+1}-\sqrt n)( \sqrt{n+1}+\sqrt n)} {\sqrt{n+1}+\sqrt n} \\ & = \frac {n+1-n} {\sqrt{n+1}+\sqrt n} \\ & = \frac 1{\sqrt{n+1}+\sqrt n} \end{aligned}$ .

$X$ is the largest when $n=1$ . Therefore, $X \le \dfrac 1{\sqrt 2 + 1} \approx 0.4142 < \dfrac 35$ . The statement is false .

Will difference always works?

1 solution

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