SOLUTION: Show that n <= 1 +sqrt(2)+sqrt(3)+...+sqrt(n) <= n(n+1)/2

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Question 990264: Show that n <= 1 +sqrt(2)+sqrt(3)+...+sqrt(n) <= n(n+1)/2
Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
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Show that n <= 1 + sqrt%282%29 + sqrt%283%29 + . . . + sqrt%28n%29 <= %28n%28n%2B1%29%29%2F2
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It is very easy.

First,  n <= 1 + sqrt%282%29 + sqrt%283%29 + . . . + sqrt%28n%29.

Replace every  sqrt%28k%29  by the smaller quantity of  1,  and you will get this inequality.


Second,  1 + sqrt%282%29 + sqrt%283%29 + . . . + sqrt%28n%29 <= %28n%28n%2B1%29%29%2F2.

Replace every  sqrt%28k%29  by the greater quantity of  k,  and you will get this inequality.  (You will have the sum of first  n  natural numbers which is exactly  %28n%28n%2B1%29%29%2F2).