document.write( "Question 788283: i forgot the formula....find the sum of the numbers in the sequence, 8+9+10+11+...400 \n" ); document.write( "
Algebra.Com's Answer #478472 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
8+9+10+11+...400
\n" ); document.write( "
\r\n" );
document.write( "Two ways to find it, the \"thinking\" way, and the \"robotic\" way:\r\n" );
document.write( "\r\n" );
document.write( "Method 1.  By writing the series in reverse order and adding, a method\r\n" );
document.write( "           in which you understand why it works.\r\n" );
document.write( "\r\n" );
document.write( "Method 2.  By using two memorized formulas that you have no idea\r\n" );
document.write( "           why they work.\r\n" );
document.write( "\r\n" );
document.write( "Method 1:\r\n" );
document.write( "\r\n" );
document.write( "8 + 9 + 10 + 11 + ... 397 + 398 + 399 + 400\r\n" );
document.write( "\r\n" );
document.write( " 8 is term #1, and 1 is 7 less than  8\r\n" );
document.write( " 9 is term #2, and 2 is 7 less than  9\r\n" );
document.write( "10 is term #3, and 3 is 7 less than 10\r\n" );
document.write( "11 is term #4, and 4 is 7 less than 11\r\n" );
document.write( "\r\n" );
document.write( "Therefore we conclude that 400 is term #393 \r\n" );
document.write( "because 393 is 7 less than 400.\r\n" );
document.write( "\r\n" );
document.write( "So there are 393 terms.\r\n" );
document.write( "\r\n" );
document.write( "Write the sequence and underneath write the sequence in\r\n" );
document.write( "reverse order:\r\n" );
document.write( "\r\n" );
document.write( "  Sum =   8 +   9 +  10 +  11 + ... + 397 + 398 + 399 + 400\r\n" );
document.write( "  Sum = 400 + 399 + 398 + 397 + ... +  11 +  10 +   9 +   8\r\n" );
document.write( "\r\n" );
document.write( "Now draw a line underneath and add them term by term:\r\n" );
document.write( "\r\n" );
document.write( "  Sum =   8 +   9 +  10 +  11 + ... + 397 + 398 + 399 + 400\r\n" );
document.write( "  Sum = 400 + 399 + 398 + 397 + ... +  11 +  10 +   9 +   8\r\n" );
document.write( "---------------------------------------------------\r\n" );
document.write( "2×Sum = 408 + 408 + 408 + 408 + ... + 408 + 408 + 408 + 408\r\n" );
document.write( "\r\n" );
document.write( "So we have 393 terms each of which is 408.\r\n" );
document.write( "\r\n" );
document.write( "2×Sum = 393×408 = 160344\r\n" );
document.write( "\r\n" );
document.write( "Therefore the sum is half of that\r\n" );
document.write( "\r\n" );
document.write( "  Sum = \"1%2F2\"×156264 = 80172\r\n" );
document.write( "\r\n" );
document.write( "=======================================================\r\n" );
document.write( "\r\n" );
document.write( "Method 2:\r\n" );
document.write( "\r\n" );
document.write( "Memorize these two formula which you have no idea why they works.\r\n" );
document.write( "\r\n" );
document.write( "an = a1 + (n-1)d\r\n" );
document.write( "\r\n" );
document.write( "Sn = \"n%2F2\"(a1 + an)\r\n" );
document.write( "\r\n" );
document.write( "where a1 = first term = 8\r\n" );
document.write( "n = the number of terms\r\n" );
document.write( "d = the difference between any term and the one before it = 1 \r\n" );
document.write( "an = the nth term\r\n" );
document.write( "Sn = the sum of the first n terms.\r\n" );
document.write( "\r\n" );
document.write( "We use the first memorized formula to find n:\r\n" );
document.write( "\r\n" );
document.write( "an = a1 + (n-1)d\r\n" );
document.write( "400 = 8 + (n-1)(1)\r\n" );
document.write( "400 = 8 + n - 1\r\n" );
document.write( "400 = 7 + n\r\n" );
document.write( "393 = n\r\n" );
document.write( "\r\n" );
document.write( "Then we substitute in the sum formula we have memorized:\r\n" );
document.write( "\r\n" );
document.write( "Sn = \"n%2F2\"(a1 + an)\r\n" );
document.write( "S393 = \"393%2F2\"(8 + 400)\r\n" );
document.write( "S393 = \"393%2F2\"(408)\r\n" );
document.write( "S393 = 80172 \r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );