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Prove the following by using the principle of mathematical induction for all $n \in N$ \[\] $1.2+2.2^2+3.2^2+...+n.2^n=(n-1)2^{n+1}+2$

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Let the given statement be $P(n)$, i.e.,
$P(n) : 1.2+2.2^2+3.2^2+...+n.2^n=(n-1)2^{n+1}+2$
For $n = 1$, we have
$P(1) : 1.2 = 2 =(1-1)2^{1+1}+2=0+2=2$, which is true.
Let $P(k)$ be true for some positive integer $k$, i.e.,
$ 1.2+2.2^2+3.2^2+...+k.2^k=(k-1)2^{k+1}+2$------------(i)
We shall now prove that $P(k+1)$ is true.
$\{1.2+2.2^2+3.2^3+...+k.2^k \}+(k+1).2^{k+1}$
$ = 2^{k+1} \{ (k-1)+(k+1) \}+2$
$ \{ (k+1)-1 \} 2^{(k+1)+1}+2$
Thus, $P(k+1)$ is true whenever $P(k)$ is true.
Hence, by the principle of mathematical induction, statement $P(n)$ is true for all natural numbers i.e., $n$.
answered Apr 29, 2014 by thanvigandhi_1

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