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, ...
I don't see a pattern.
How about this?
1, 16, 81, 256, 625, ....
= 1^4, 2^4, 3^4, 4^4, 5^4, ...
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While I agree tutor greenstamps came up with the correct answer, there is no basis for assuming the next dif4 will be 24. There is not enough data to make that calculation/assumption.
I will borrow the beginning of my response from the other tutor....
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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, ...
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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.
Let's make the common difference on this row of dif4:
1, 16, 81, 256, 625, ....
dif 15, 65, 175, 369, ...
dif2 50, 110, 194, ...
dif3 60, 84, ...
dif4 24, 24, ...
Now we can work back up the table to find the next term in the sequence.
1, 16, 81, 256, 625, 1296 ....
dif 15, 65, 175, 369, 671 ...
dif2 50, 110, 194, 302 ...
dif3 60, 84, 108, ...
dif4 24, 24, ...
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.