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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.

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