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Successive differences of powers

Successive Differences of Powers

Write down the square series

0, 1, 4, 9, 16, 25, 36, 49, 64...

Take the successive difference of the square series

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21...

Write down the difference of second successive series, that is,

2, 2, 2, 2, 2, 2, 2....

Here the second successive difference are constant and equal to 2.

Write down the cube series as done before for the squares

0, 1, 8, 27, 64, 125, 216, 343, 512...

Write their successive difference

1, 7, 19, 37, 61, 91, 127, 169, ...

6, 12, 18, 24, 30, 36, 42,...

6, 6, 6, 6, 6, 6,...

Here the third successive difference are all constant and equal to 6.

This process is same for all the successive difference of 4th, 5th, 6th...powers. This process has practical application to find the nth order derivative with the first N terms of the polynomial to solve the generating function.

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