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:$$\begin{align*}
\sum_{k=1}^nk2^k&=\sum\nolimits_1^{n+1} x2^x\,\delta x\\&=\sum\nolimits_1^{n+1} x\Delta2^x\,\delta x\\&=\left[x2^x\right]_1^{n+1}-\sum\nolimits_1^{n+1} 2^{x+1}\Delta x\,\delta x\\&=(n+1)2^{n+1}-2-\sum\nolimits_1^{n+1} 2^{x+1}\,\delta x\\
&=(n+1)2^{n+1}-2-[2^{x+1}]_1^{n+1}\\
&=(n+1)2^{n+1}-2-(2^{n+2}-4)\\