document.write( "Question 310266: Please help me solve this...
\n" ); document.write( "write the following in summation notation
\n" ); document.write( "1+6+13+22... \n" ); document.write( "
Algebra.Com's Answer #221861 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
Please help me solve this...
\n" ); document.write( "write the following in summation notation
\n" ); document.write( "1+6+13+22...
\n" ); document.write( "
\r\n" );
document.write( "I assume you want it in the form \"sum%28a%5Bn%5D%2C+n=1%2C+infinity%29\"\r\n" );
document.write( "\r\n" );
document.write( "First we find the general term by assuming its general term\r\n" );
document.write( "can be expressed as a polynomial.  To find the smallest\r\n" );
document.write( "possible degree it can have we write the four given numbers in\r\n" );
document.write( "a line like this\r\n" );
document.write( "\r\n" );
document.write( "1   6   13   22 \r\n" );
document.write( "\r\n" );
document.write( "Then we find a line of differences by subtracting each number \r\n" );
document.write( "from the one just to the right of it and writing the difference \r\n" );
document.write( "between and below the numbers subtracted, like this:\r\n" );
document.write( "\r\n" );
document.write( "1   6   13   22\r\n" );
document.write( "  5   7    9\r\n" );
document.write( "\r\n" );
document.write( "5, 7, and 9 are not all the same number, so we do the same to \r\n" );
document.write( "the bottom line:\r\n" );
document.write( "\r\n" );
document.write( "1   6   13   22\r\n" );
document.write( "  5   7    9\r\n" );
document.write( "    2    2\r\n" );
document.write( "\r\n" );
document.write( "Those are the same, since they are both 2's.  Since it requires 2 lines\r\n" );
document.write( "of differences to get them all to be the same number, we assume the general\r\n" );
document.write( "term is a polynomial of degree 2:\r\n" );
document.write( "\r\n" );
document.write( "\"a%5Bn%5D=An%5E2%2BBn%2BC\"\r\n" );
document.write( "\r\n" );
document.write( "We substitute n=1,2, and 3\r\n" );
document.write( "\r\n" );
document.write( "  \r\n" );
document.write( "\r\n" );
document.write( "\"system%281=A%2BB%2BC%2C%0D%0A6=4A%2B2B%2BC%2C%0D%0A13=9A%2B3B3%2BC%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"system%28A%2BB%2BC=1%2C%0D%0A4A%2B2B%2BC=6%2C%0D%0A9A%2B3B3%2BC=13%29\"\r\n" );
document.write( "\r\n" );
document.write( "We solve that system and get\r\n" );
document.write( "\r\n" );
document.write( "\"A=1\", \"B=2\", and \"C=-2\"\r\n" );
document.write( "\r\n" );
document.write( "So,\r\n" );
document.write( "\r\n" );
document.write( "\"a%5Bn%5D=1n%5E2%2B2n-2\"\r\n" );
document.write( "\r\n" );
document.write( "\"a%5Bn%5D=n%5E2%2B2n-2\"\r\n" );
document.write( "\r\n" );
document.write( "So we write:\r\n" );
document.write( "\r\n" );
document.write( "\"sum%28a%5Bn%5D%2C+n=1%2C+infinity%29+=+sum%28+%28n%5E2%2B2n-2%29%2C+n=1%2C+infinity+%29\"\r\n" );
document.write( "\r\n" );
document.write( "Note: The series could also be written in the form:\r\n" );
document.write( "\r\n" );
document.write( "\"sum%28a%5Bn%5D%2C+n=0%2C+infinity%29+=+sum%28+%28n%5E2%2B4n%2B1%29%2C+n=0%2C+infinity+%29\"\r\n" );
document.write( "\r\n" );
document.write( "where the index (or \"dummy variable\") n starts at 0.  Then the\r\n" );
document.write( "system of equations would be easier to solve.  But I am guessing \r\n" );
document.write( "that your teacher probably wanted the summation index to start at\r\n" );
document.write( "1, not 0.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

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