Why do these patterns occur in powers

noetsi

No cake for spunky
#1
A strange question but something I have wondered for a long time.

If you look at the differences in squares there is an obvious pattern.

o
1 3
4 5
9 7
16 9
25

The differences go on like this forever as far as I know. That is the differences between the squares increase by the next odd number 3 5 7 9 11 15....

There is a similar pattern with cubes

0
1 1
8 7 6
27 19 12
64 37 18
125 61 24

The differences between the differences of the cubes go up in a patter of 6's 6,12, 18,24 etc.

There is a pattern for fourth powers as well and I assume more advanced powers.

Anyone know what causes this? I am sure it's not concidental.
 

Dason

Ambassador to the humans
#2
Well it boils down to the fact that: (x+1)^p - x^p reduces down to a polynomial with order p-1 instead of p.

For example in the p = 2 case we have: (x+1)^2 - x^2 = (x^2 + 2x + 1) - x^2 = 2x + 1.

Does this work? If we let x = 3 then we're looking at 4^2 - 3^2 = 16 - 9 = 7. This is the same as 7 = 2*3+1
 

noetsi

No cake for spunky
#3
Yes. I always assumed it must be something along that line but I did not know the details. Thanks for the comment Dason.