an = (n-1/2).2^n
= 2(n.2^(n-1)) - 2^(n-1)
consider
1+x+x^2+..+x^n= (x^(n+1) -1)/(x-1)
1+2x+..+nx^(n-1)
=[(x^(n+1) -1)/(x-1)]'
={nx^(n+1)-(n+1)x^n +1 } /(x-1)^2
put x=2
1.2^0+2.2^1+...+n.2^(n-1) = n2^(n+1)-(n+1)2^n +1
an = 2(n.2^(n-1)) - 2^(n-1)
Sn = a1+a2+..+an
= 2{n2^(n+1)-(n+1)2^n +1} - (2^n-1)
= (2n-3).2^n +3
sn/2^n = (2n-3) + 3/2^n
> 2n-3