Sigma (k=1...2²-1 )=Sigma ( k=1...2²-1 )+Sigma ( k=2²...3²-1 )+....+Sigma (k=(n-1)² ... n²-1 )
=Sigma (p=2 ... n)Sigma (k=(p-1)² ... p²-1 )
si k=(p-1)²=<k=<p²-1 ==> p-1=<racin( k )<p
===> E( racin( k ) )=p-1
==> Sigma (k=(p-1)² ... p²-1 )E( racin( k ) )=(p-1)(p²-(p-1)² )=(p-1)(2p-1)=2p²-3p+1
Sn=Sigma (p=2 ... n)(2p²-3p+1)
=2Sigma (p=2 ... n)p²-3Sigma (p=2 ... n)p+n-1
=2(-1+n(n+1)(2n+1)/6)-3(-1+n(n+1)/2)+n-1
_________________
وقل ربي زد ني علما