2024/055) if $2n+ 1$ is a perferct square show that $n+1$ is sum of 2 squares when n is an integer

 We know $2n+1$ is odd so it is square of an odd number say $(2k+1)^2$

So $2n+1 = (2k+1)^2 = 4k^2 + 4k +1$

or $2n= 4k^2+ 4k$

or $n = 2k^2 + 2k$

or $n + 1=  2k^2+2k + 1 = k^2+(k^2+2k+1) = k^2+(k+1)^2$

Proved