Question

1+2x2+3x2^2+4x2^3+.+nx2^n-1=?

Answer 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