Problem of the week #85 - November 11th, 2013

[Jameson]

Which is larger? $\sqrt{7}-\sqrt{6}$ or $\sqrt{6}-\sqrt{5}$? Answer without using a calculator.
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[Jameson]

Congratulation to the following members for their correct solutions:

1) MarkFL
2) anemone
3) Pranav
4) eddybob123
5) kaliprasad

Solution (from eddybob123):

Spoiler

$$ 5041 > 5040$$
$$\Rightarrow 71>2\sqrt{1260}$$
$$\Rightarrow 1< 72-2\sqrt{1260}$$
$$\Rightarrow 1^2< (\sqrt{42}-\sqrt{30})^2$$
$$\Rightarrow 1< \sqrt{42}-\sqrt{30}$$
$$\Rightarrow 2<2(\sqrt{42}-\sqrt{30})$$
$$\Rightarrow 7-2\sqrt{42}+6<6-2\sqrt{30}+5$$
$$\Rightarrow (\sqrt{7}-\sqrt{6})^2<(\sqrt{6}-\sqrt{5})^2$$
$$\Rightarrow \sqrt{7}-\sqrt{6}<\sqrt{6}-\sqrt{5}$$