[tex](2n+1)^2=4n^2+4n+1=4n(n+1)+1[/tex]
[tex]n(n+1)[/tex] is a product of two consecutive numbers, so it's divisible by 2. Therefore, the product [tex]4n(n+1)[/tex] is divisible by [tex]4\cdot2=8[/tex]. In other words, that product is a multiple of 8. So [tex]4n(n+1)+1[/tex] is always "1 more that a multiple of 8".
