Question

3²=4+5 5²=12+13 7²=24+25 有什么规律

Answer 1

应该是(2n-1)²+{[(2n-1)²-1]/2}²={[(2n-1)²+1]/2}²

证明
{[(2n-1)²+1]/2}²-{[(2n-1)²-1]/2}²
平方差
={[(2n-1)²+1]/2+[(2n-1)²-1]/2}{[(2n-1)²+1]/2-[(2n-1)²-1]/2}
=(2n-1)²×1
=(2n-1)²
所以(2n-1)²+{[(2n-1)²-1]/2}²={[(2n-1)²+1]/2}²

Answer 2

（2n+1）^2=(n(2n+1)+n)+(n(2n+1)+n+1)

追问

如何用勾股知识证明