We build a chart
color(white)(aaaa)Xcolor(white)(aaaa)1color(white)(aaaa)2color(white)(aaaaaa)3color(white)(aaaa)..........color(white)(aaaa)(n-1)color(white)(aaaa)n
color(white)(aaaa)pcolor(white)(aaaa)1/ncolor(white)(aaaa)1/ncolor(white)(aaaa)1/ncolor(white)(aaaa)..........color(white)(aaaaaa)1/ncolor(white)(aaaaa)1/n
color(white)(aaaa)Xpcolor(white)(aaa)1/ncolor(white)(aaaa)2/ncolor(white)(aaaa)3/ncolor(white)(aaaa)..........color(white)(aaaa)(n-1)/ncolor(white)(aaaa)n/n
color(white)(aaaa)X^2pcolor(white)(aa)1^2/ncolor(white)(aaaa)2^2/ncolor(white)(aaaa)3^2/ncolor(white)(aaaa)..........color(white)(a)(n-1)^2/ncolor(white)(aaaa)n^2/n
The Expected value is
E(X)=sum_1^nXp=1/n(1+2+3+...+(n-1)+n)
=1/n*n/2(n+1)
=1/2(n+1)
sum_1^nX^2p=1/n(1^2+2^2+3^2+........+(n-1)^2+n^2)
=1/n*n/6(n+1)(2n+1)
=1/6(n+1)(2n+1)
The variance is
var(X)=(sum_1^nX^2p)-(E(X))^2
=1/6(n+1)(2n+1)-1/4(n+1)^2
=1/2(n+1)(1/3(2n+1)-1/2(n+1))
=1/2(n+1)*1/6(4n+2-3n-3)
=1/12(n+1)(n-1)
=1/12(n^2-1)
QED
I hope that this is clearer.