Đặt `X = 2 . 2^2+3 . 2^3+...+n . 2^n `
` 2X = 2 . 2^3 + 3 . 2^4 + .... + n . 2^(n+1)`
` 2X - X = (2 . 2^3 + 3 . 2^4 + ... + n . 2^(n+1) ) - (2 . 2^2 + 3 . 2^3 + ... + n . 2^n)`
` X = n . 2^(n+1) - 2^3 - (2^3+2^4+...+2^n)`
Đặt `Y = 2^3+2^4+...+2^n`
` 2Y = 2^4+2^5+...+2^(n+1)`
` 2Y - Y = (2^4+2^5+...+2^(n+1) ) - (2^3+2^4+...+2^n)`
` Y = 2^(n+1) - 2^3`
` => X = n . 2^(n+1) - 2^3 - 2^(n+1) + 2^3`
` => X = n . 2^(n+1) - 2^(n+1)`
` => 2^(n+11) = 2^(n+1) . (n-1)`
` => n-1 = 2^(n+11) : 2^(n+1)`
` => n-1 = 2^10`
` => n - 1 = 1024`
` n = 1024+1`
` n = 1025`
Vậy `n=1025`
