document.write( "Question 990264: Show that n <= 1 +sqrt(2)+sqrt(3)+...+sqrt(n) <= n(n+1)/2 \n" ); document.write( "

Algebra.Com's Answer #610360 by ikleyn(52800) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Show that n <= 1 + $\"sqrt%282%29\"$ + $\"sqrt%283%29\"$ + . . . + $\"sqrt%28n%29\"$ <= $\"%28n%28n%2B1%29%29%2F2\"$
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\n" ); document.write( "\n" ); document.write( "It is very easy.\r
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\n" ); document.write( "\n" ); document.write( "First, n <= 1 + $\"sqrt%282%29\"$ + $\"sqrt%283%29\"$ + . . . + $\"sqrt%28n%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Replace every $\"sqrt%28k%29\"$ by the smaller quantity of 1, and you will get this inequality.\r
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\n" ); document.write( "\n" ); document.write( "Second, 1 + $\"sqrt%282%29\"$ + $\"sqrt%283%29\"$ + . . . + $\"sqrt%28n%29\"$ <= $\"%28n%28n%2B1%29%29%2F2\"$ . \r
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\n" ); document.write( "\n" ); document.write( "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 $\"%28n%28n%2B1%29%29%2F2\"$ ). \r
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