Successive Differences of Powers

List the squares:

0, 1, 4, 9, 16, 25, 36, 49, …

Then take their successive differences:

1, 3, 5, 7, 9, 11, 13, …

Then take their successive differences again:

2, 2, 2, 2, 2, 2, …

So the 2nd successive differences are constant(!) and equal to 2.

OK, now list the cubes, and in a similar way, keep taking successive differences:

0, 1, 8, 27, 64, 125, 216, 343, 512, … 
1, 7, 19, 37, 61, 91, 127, 169, … 
6, 12, 18, 24, 30, 36, 42, … 
6, 6, 6, 6, 6, 6, …

Gee, the 3rd successive differences are all constant(!) and equal to 6. 
What happens when you take the 4th successive differences of 4th powers? Are they constant? What do they equal? (They’re all 24.) And the 5th successive differences of 5th powers?

Aren’t derivatives similar to differences? What do you think happens when you take the n-th derivative of xn?

Presentation Suggestions:
Have students do these investigations along with you. If you assign the n-th derivative of xn on a previous homework, then you can make the connection between the two right away.

The Math Behind the Fact:
This pattern may seem very surprising. It can be proved by induction. Taking differences is like a discrete version of taking the derivative, where the space between successive points is 1.

This idea has a very practical application: given a sequence generated by an unknown polynomial function, you use the calculation of successive differences to determine the order of the polynomial! Then use the first N terms of the sequence with the first N terms of the polynomial to solve for the generating function.

How to Cite this Page: 
Su, Francis E., et al. “Successive Differences of Powers.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Francis Su

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