Question 1142531
I will borrow the beginning of my response from the other tutor....<br>
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I will list the differences and then differences-of-differences, etc.

       1,   16,   81,    256,    625,   ....
dif      15,   65,    175,   369,   ...
dif2        50,   110,    194,  ...
dif3           60,     84,   ...
dif4               24,   ...</pre>
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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.<br>
Let's make the common difference on this row of dif4:<br><pre>
       1,   16,   81,    256,    625,   ....
dif      15,   65,    175,   369,   ...
dif2        50,   110,    194,  ...
dif3           60,     84,   ...
dif4               24,     24,   ...</pre>
Now we can work back up the table to find the next term in the sequence.<br><pre>
       1,   16,   81,    256,    625,  1296   ....
dif      15,   65,    175,   369,   671   ...
dif2        50,   110,    194,   302   ...
dif3           60,     84,   108,   ...
dif4               24,     24,   ...</pre>
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.