Answer
$384$
Work Step by Step
We calculate the given sum by expanding it: we calculate each term for $k=1,2,3,4,5,6$ and add them:
$$\begin{align*}
\sum_{k=1}^6 (k+1)2^{k-1}&=2(2^0)+3(2^1)+4(2^2)+5(2^3)+6(2^4)+7(2^5)\\
&=2+6+16+40+96+224\\
&=384.
\end{align*}$$
