What is the name or rule for this observation on squares that I've explained below?
If you're at a number, say 400 (m^2) which is a square of 20 (m), and you want to find the square of 25, that is 5 (n) numbers ahead. The square of 25 will be (2 * m * n) + n^2 ahead of m^2.
(25)^2 = 400 + 2 * 20 * 5 + 5^2 = 400 + 200 + 25 = 625
Another example: if you're at a number, say 121 (m^2) which is a square of 11 (m), and you want to find the square of 15, that is 4 (n) numbers ahead.
(15)^2 = 121 + 2 * 11 * 4 + )^2 = 121 + 88 + 16 = 225
This observation will work very well within an algorithm/program where you need to efficiently brute force to find the decimal/float square root of any number, by finding the integer/whole-number square roots of the original number multiplied by powers of 100 in every iteration.