Prove the following by using the principle of mathematical inductio...

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Prove the following by using the principle of mathematical induction for all \( n \in \mathbf{N} \) : \( 1.2+2.2^{2}+3.2^{3}+\ldots+n .2^{n}=(n-1) 2^{n+1}+2 \).
\( \mathrm{P} \)
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