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**emakarov** In the second last line, you write $\displaystyle \displaystyle\sum_{i=1}^{k+2}i\cdot2^i=\sum_{i=1}^ {k+1}i\cdot2^i+k\cdot2^k$, whereas the right-hand side should be $\displaystyle \displaystyle\sum_{i=1}^{k+1}i\cdot2^i+(k+2)\cdot2 ^{k+2}$.

A style remark. In the beginning of the induction step, you write, "If p(k) is true for all integers k >= 2, then p(k+1) is true". First, it's helpful to write if this is your goal to prove or if you have already shown it. Second, this sentence can be interpreted as $\displaystyle (\forall k\ge 2.\;p(k))\to p(k+1)$, which is trivial. It is better to say, "For all k >= 2, if p(k), then p(k+1)" or "p(k) implies p(k+1) for all k >= 2".