1+2×2+3×2^2+4×2^3+...+n×2^(n-1)
=2^0+2^1+2^2+2^3+...+2^(n-1)
+2^1+2^2+2^3+...+2^(n-1)
+2^2+2^3+...+2^(n-1)
+2^3+...+2^(n-1)
+...+...
+2^(n-1)
=(2^n-2^0)+(2^n-2^1)+(2^n-2^2)+(2^n-2^3)+...+[2^n-2^(n-1)]
=n2^n-(2^n-2^0)
=(n-1)2^n+1
1+2×2+3×2^2+4×2^3+...+n×2^(n-1)
=2^0+2^1+2^2+2^3+...+2^(n-1)
+2^1+2^2+2^3+...+2^(n-1)
+2^2+2^3+...+2^(n-1)
+2^3+...+2^(n-1)
+...+...
+2^(n-1)
=(2^n-2^0)+(2^n-2^1)+(2^n-2^2)+(2^n-2^3)+...+[2^n-2^(n-1)]
=n2^n-(2^n-2^0)
=(n-1)2^n+1