## Sunday, May 17, 2015

### Show with proof which of these two values is smaller: $7$, or $\sqrt 2+\sqrt 5 + \sqrt {11}$

Show with proof which of these two values is smaller:

$7$, or $\sqrt 2+\sqrt 5 + \sqrt {11}$

I know this problem could be solved using numerous different methods but I am going to post my solution anyway, since I think and I believe I have done a great job in proving it using the most elementary method and the best of observation and thus it earned a place at this blog. :D

My solution:

Observe that

$288<289\,\,\implies2(12^2)<17^2$ or $(\sqrt{2}<\dfrac{17}{12})---(1)$

$80<81\,\,\implies5(4^2)<9^2$ or $(\sqrt{5}<\dfrac{9}{4})---(2)$

$99<100\,\,\implies11(3^2)<10^2$ or $(\sqrt{11}<\dfrac{10}{3})---(3)$

Adding the inequalities in (2), (2) and (3) up gives us the answer:

$\sqrt{2}+\sqrt{5}+\sqrt{11}<\dfrac{17}{12}+\dfrac{9}{4}+\dfrac{10}{3}=7$

Therefore, $\sqrt 2+\sqrt 5 + \sqrt {11}$ is smaller than $7$.