sorry for every possibile grammatical mistake I may make, anyway, I’d like to ask you if this rule has been “discovered” before: A square number is formed by the sum of the previous square number and the corresponsing odd number so:
25(5th square number)=16(previous square number)+9(5th odd number)
64(8th square number)=49(previous square number)+15(8th odd number)
4(2nd square number)=1(previous square number)+3(2nd odd number)
1(first square number)=1(because there’s no previous square number and 1 is the first odd number)
Eccetera eccetera. Has this ever been discovered?
Let $x^2$ be the xth square number. The xth odd number is just 2x - 1, and the previous square is (x - 1)^2. Thus, you're showing that:
$$x^2 = (x - 1)^2 + 2x - 1$$ $$x^2 = x^2 - 2x + 1 + 2x - 1$$ $$x^2 = x^2$$