document.write( "Question 1209378: What is the 1000th value of the following sequence?
\n" ); document.write( "1/1,1/2,2/2,1/3,2/3,3/3,1/4,2/4,3/4,4/4,... \n" ); document.write( "
Algebra.Com's Answer #848709 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "\r\n" );
document.write( "1/1,\r\n" );
document.write( "1/2,2/2,\r\n" );
document.write( "1/3,2/3,3/3,\r\n" );
document.write( "1/4,2/4,3/4,4/4,\r\n" );
document.write( "...\r\n" );
document.write( "\r\n" );
document.write( "The 1st 1 is term number 1=1 and it is written 1/1\r\n" );
document.write( "The 2nd 1 is term number 1+2=3 and it is written 2/2\r\n" );
document.write( "The 3rd 1 is term number 1+2+3=6 and it is written 3/3\r\n" );
document.write( "The 4th 1 is term number 1+2+3+4=10 and it is written 4/4\r\n" );
document.write( "\r\n" );
document.write( "The sum of the first k positive integers is \"k%28k%2B1%29%2F2\"\r\n" );
document.write( "\r\n" );
document.write( "So\r\n" );
document.write( "\r\n" );
document.write( "The nth 1 is term number \"%28n%28n%2B1%29%29%2F2\" and it is written n/n\r\n" );
document.write( "\r\n" );
document.write( "Let's find the term number of the last 1 before the 1000th term\r\n" );
document.write( "\r\n" );
document.write( "\"%28n%28n%2B1%29%29%2F2+%3C=+1000\"\r\n" );
document.write( "\"n%28n%2B1%29+%3C=+2000\"\r\n" );
document.write( "\"n%5E2%2Bn+%3C=+2000\"\r\n" );
document.write( "\"n%5E2%2Bn-2000%3C=0\"\r\n" );
document.write( "Find the positive zero of the left side by the quadratic formula\r\n" );
document.write( "and get approximately 44.224\r\n" );
document.write( "\r\n" );
document.write( "So we substitute 44 in  \r\n" );
document.write( "\"%28n%28n%2B1%29%29%2F2\" \r\n" );
document.write( "\"%2844%2A45%29%2F2\"\r\n" );
document.write( "\"990\"\r\n" );
document.write( "\r\n" );
document.write( "So the last 1 before the 1000th term is the 990th term, written as 44/44.\r\n" );
document.write( "\r\n" );
document.write( "So we either think of, or write out the next 10 terms.\r\n" );
document.write( "\r\n" );
document.write( "1/45,2/45,3/45,4/45,5/45,6/45,7/45,8/45,9/45,10/45\r\n" );
document.write( "\r\n" );
document.write( "So the 1000th term is 10/45.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( " \n" ); document.write( "
\n" );