# Is there a name for this pattern involving the differences between perfect squares, cubes, etc?

I stumbled upon this pattern and was wondering if it has a name or any applications?

Take a set of consecutive perfect squares and find the difference between any consecutive pair. Then find the difference between each consecutive difference equals 2.

• 4 - 1 = 3
• 9 - 4 = 5 -> 5-3 = 2
• 16 - 9 = 7 -> 7-5 = 2
• 25 - 16 = 9 -> 9-7 = 2

Take a set of consecutive perfect cubes and find the difference between any consecutive pair. Then find the difference between the difference between each consecutive difference equals 6.

• 8 – 1 = 7
• 27 – 8 = 19 -> 19 - 7 = 12
• 64 – 27 = 37 -> 37 - 19 = 18 -> 18-12 = 6
• 125 – 64 = 61 -> 61 = 37 = 24 -> 24 -18 = 6

The same applies for x^4 but as you may suspect you must use another “layer of differences” before finding the common difference of 24.

Screenshot of spreadsheet with additional examples:

https://i.stack.imgur.com/ksJjD.png

• This is the "calculus of finite differences" – Lord Shark the Unknown Dec 5 '17 at 2:45

## 1 Answer

You are hitting on an idea of calculus. What you are examining is the discrete derivatives of $x^n$. The n-th iteration will give you $n!$.

• Very interesting, thanks! – Brian Dec 5 '17 at 3:10