2.2^2+3.2^3+4.2^4+…+n.2^n=2^n+34 Ai giải nhanh, đúng ctrlhn+5 sao

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1. Đáp án:`n=2^{33}+1`

   Giải thích các bước giải:

   Đặt `S=2.2^2+3.2^3+4.2^4+…+n.2^`

   `=>2S=2(2.2^2+3.2^3+4.2^4+…+n.2^n)`

   `=>2S=2.2^3+3.2^4+4.2^5+…+n.2^{n+1}`

   `=>2S-S=S=2.2^2+(3.2^3-2.2^3)+…+(n-n+1).2^n-n.2^{n+1}`

   `=>S=2.2^2+2^3+2^4+…+2^n-n.2^{n+1}`

   `=>S=2^2+(2^2+2^3+…+2^{n+1})-\(n+1).2^{n+1}`

   `=>S=-2^2-(2^2+2^3+…+2^{n+1})+(n+1).2^{n+1}`

   Đặt `K=2^2+2^3+…+2^{n+1}`

   `=>2K=2^3+2^4+…+2^{n+2}`

   `=>2K-K=K=2^{n+2}-2^2`

   `=>S=2^2-2^{n+2}+2^2+(n+1).2^{n+1}`

   `=>S=(n+1).2^{n+1}-2^{n+2}`

   `=>S=2^{n+1}(n-1)`

   Mà `S=2^{n+34}`

   `=>n-1=(2^{n+34})/(2^{n+1})`

   `=>n-1=2^{33}`

   `=>n=2^{33}+1`

   Vậy `n=2^{33}+1`

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