document.write( "Question 1086342: The sum of the ﬁrst n integers 1 + 2 + 3 + ... + n =
\n" ); document.write( "n(n + 1)/2 a How many numbers must be taken to have a sum greater than one million? b Why can’t the sum ever equal 100 000?
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Algebra.Com's Answer #700509 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"%28n%28n%2B1%29%29%2F2%3E1000000\"$
\n" ); document.write( " $\"n%28n%2B1%29%3E2000000\"$
\n" ); document.write( " $\"n%5E2%2Bn%3E2000000\"$
\n" ); document.write( " $\"n%5E2%2Bn-2000000%3E0\"$
\n" ); document.write( "We can solve for when it equals zero and then take the next largest integer,
\n" ); document.write( " $\"n%5E2%2Bn-2000000=0\"$
\n" ); document.write( " $\"%28n%5E2%2Bn%2B1%2F4%29=2000000%2B1%2F4\"$
\n" ); document.write( " $\"%28n%2B1%2F2%29%5E2=8000001%2F4\"$
\n" ); document.write( " $\"n%2B1%2F2=0+%2B-+sqrt%288000001%29%2F2\"$
\n" ); document.write( " $\"n=1%2F2+%2B-+%283sqrt%28888889%29%29%2F2\"$
\n" ); document.write( "Only positive n makes sense in this problem,
\n" ); document.write( " $\"n=%281%2B3sqrt%28888889%29%29%2F2\"$
\n" ); document.write( "or approximately,
\n" ); document.write( " $\"n=1413.71\"$
\n" ); document.write( "So the next largest integer is,
\n" ); document.write( " $\"N=1414\"$
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\n" ); document.write( "Assume the sum does equal 100000,
\n" ); document.write( " $\"%28n%28n%2B1%29%29%2F2=10000\"$
\n" ); document.write( " $\"n%5E2%2Bn=200000\"$
\n" ); document.write( " $\"n%5E2%2Bn-200000=0\"$
\n" ); document.write( "Similarly,
\n" ); document.write( " $\"%28n%2B1%2F2%29%5E2=200000%2B1%2F4\"$
\n" ); document.write( " $\"%28n%2B1%2F2%29%5E2=800001%2F4\"$
\n" ); document.write( "Since the $\"sqrt%28800001%29\"$ is not an integer, there is no hope that the required sum would be an integer.
\n" ); document.write( "So it would never equal 100000.
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