document.write( "Question 934953: 4/n^4+5/n^4+6/n^4+...+(n^4-4)/n^4+(n^4-5)/n^4+(n^4-6)/n^4=309? \n" ); document.write( "

Algebra.Com's Answer #568514 by KMST(5328) $\"\"$ $\"About$

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Rearranging that sum so that each numerator is greater that the one before, I get
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\n" ); document.write( " $\"%284%2B5%2B6%2B%22...%22%2B%28n%5E4-6%29%2B%28n%5E4-5%29%2B%28n%5E4-4%29%29%2Fn%5E4=309\"$
\n" ); document.write( "The numerator in the expression above is the sum of $\"n%5E4-7\"$ terms of an arithmetic sequence with common difference $\"1\"$ ,
\n" ); document.write( "first term $\"4\"$ , and last term $\"n%5E4-4\"$ .
\n" ); document.write( "The sum of a number of terms of an arithmetic sequence is
\n" ); document.write( "the average of first and last terms
\n" ); document.write( "times the number of terms.
\n" ); document.write( "In this case, the sum is
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,
\n" ); document.write( "so we can re-write the equation as
\n" ); document.write( " $\"%28n%5E4%28n%5E4-7%29%2F2%29%2Fn%5E4=309\"$ , which simplifies to
\n" ); document.write( " $\"%28n%5E4-7%29%2F2=309\"$
\n" ); document.write( "Solving for a positive integer $\"n\"$ ,
\n" ); document.write( " $\"%28n%5E4-7%29%2F2=309\"$ ---> $\"n%5E4-7=2%2A309\"$ ---> $\"n%5E4-7=618\"$ ---> $\"n%5E4=7%2B618\"$ ---> $\"n%5E4=625\"$ ---> $\"highlight%28n=5%29\"$ .
\n" ); document.write( "If $\"n\"$ did not need to be positive, $\"n=-5\"$ would also be a solution. \n" ); document.write( "

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