[tex]s=\frac{1}{2}n(n+1) \\ \\
8s+1=8 \times \frac{1}{2}n(n+1)+1=4n(n+1)+1=4n^2+4n+1=(2n+1)^2[/tex]
n is an integer, so 2n+1 is also an integer. 8s+1 is equal to the square of 2n+1, so it's a square number.
2n+1 is an odd integer for any value of n. The square of an odd integer is always an odd integer. Therefore 8s+1=(2n+1)² is odd.
8s+1 is an odd square number.
