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\begin{align*} \sum_{i=1}^{k+1} &= (\sum_{i=1}^k i2^{i-1}) + (k+1)2^{(k+1)-1}\\ &= ((k-1) \cdot 2^k + 1) + (k+1)2^k \; \; \; \; \mathrm{(inductive \; hypothesis)}\\ &= ((k-1) + (k+1))2^k + 1 \\ &= (k \cdot 2) \cdot 2^k + 1\\ &= k2^{k+1} +1 \end{align*}