Prove the following by the principle of mathematical induction: `\ 1. 2+2. 2^2+3. 2^3++n .2^n=(n-1)2^(n+1)+2`

Explanation

`P(n):1.2+2.2^2+3.2^3+....+n.2^n=(n−1)2^(n+1)+2`

For `n=1,LHS=1.2=2`


RHS`=(1−1)2^(1+1)+2=2`

∴LHS=RHS

∴P(1) is true

Let us assume P(k) is true for some k∈N

i.e,
`1.2+2.2^2+3.2^3+...+k.2^k=(k−1)2^(k+1)+2`

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