document.write( "Question 1142531: Please help me find the common difference of the sequence 1,16,81,256,625,_ So that i can calculate for the 6th term \n" ); document.write( "
Algebra.Com's Answer #763223 by greenestamps(13200)\"\" \"About 
You can put this solution on YOUR website!
I will borrow the beginning of my response from the other tutor....

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document.write( "I will list the differences and then differences-of-differences, etc.\r\n" );
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document.write( "       1,   16,   81,    256,    625,   ....\r\n" );
document.write( "dif      15,   65,    175,   369,   ...\r\n" );
document.write( "dif2        50,   110,    194,  ...\r\n" );
document.write( "dif3           60,     84,   ...\r\n" );
document.write( "dif4               24,   ...

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\n" ); document.write( "To this point, there is no common difference. So we can create a common difference, either on this row of dif4 or on some subsequent row.

\n" ); document.write( "Let's make the common difference on this row of dif4:
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document.write( "       1,   16,   81,    256,    625,   ....\r\n" );
document.write( "dif      15,   65,    175,   369,   ...\r\n" );
document.write( "dif2        50,   110,    194,  ...\r\n" );
document.write( "dif3           60,     84,   ...\r\n" );
document.write( "dif4               24,     24,   ...

\n" ); document.write( "Now we can work back up the table to find the next term in the sequence.
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document.write( "       1,   16,   81,    256,    625,  1296   ....\r\n" );
document.write( "dif      15,   65,    175,   369,   671   ...\r\n" );
document.write( "dif2        50,   110,    194,   302   ...\r\n" );
document.write( "dif3           60,     84,   108,   ...\r\n" );
document.write( "dif4               24,     24,   ...

\n" ); document.write( "We know that a common difference of 24 in row dif4 means the sequence can be produced by a polynomial of degree 4 with leading coefficient 24/4! = 1. And in fact the 4th degree polynomial that produces this sequence is just P(x) = x^4.
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