\(Đặt\) \(A=2.2^2+3.2^3+4.2^4+...+n.2^n\)

\(2A=2.2^3+3.2^4+4.2^5+....+n.2^{n+1}\)

\(2A-A=2.2^3+3.2^4+4.2^5+....+n.2^{n+1}-\left(2.2^2+3.2^3+4.2^4+...+n.2^n\right)\)

\(=-2.2^2-2^3-2^4-...-2^n+n.2^{n+1}\)

\(=-2^2-\left(2^2+2^3+...+2^n\right)+n.2^{n+1}\)

\(=-2^2-\left(2^{n+1}-2^2\right)+n.2^{n+1}\)

\(=\left(n-1\right).2^{n+1}\)

=> \(\left(n-1\right).2^{n+1}=2^{n+16}=2^{n+1}.2^{15}\)

\(\Leftrightarrow n-1=2^{15}\)

\(\Leftrightarrow n=2^{15}+1\)

Đề thiếu bạn ạ!

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hok tốt

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Đặt A=2.22+3.23+4.24+...+n.2nA=2.22+3.23+4.24+...+n.2n

Ta có:

A=2.22+3.23+4.24+...+n.2nA=2.22+3.23+4.24+...+n.2n

⇒2A=2(2.22+3.23+4.24+...+n.2n)⇒2A=2(2.22+3.23+4.24+...+n.2n)

⇒2A=2.23+3.24+4.25+...+n.2n+1⇒2A=2.23+3.24+4.25+...+n.2n+1

⇒2A−A=2.22+(3.23−2.23)+...+(n−n+1).2n−n.2n+1⇒2A−A=2.22+(3.23−2.23)+...+(n−n+1).2n−n.2n+1

⇒A=2.22+23+24+...+2n−n.2n+1⇒A=2.22+23+24+...+2n−n.2n+1

⇒A=22+(22+23+...+2n+1)−(n+1).2n+1⇒A=22+(22+23+...+2n+1)−(n+1).2n+1

⇒A=−22−(22+23+...+2n+1)+(n+1).2n+1⇒A=−22−(22+23+...+2n+1)+(n+1).2n+1

Đặt B=22+23+...+2n+1B=22+23+...+2n+1

⇒2B=23+24+...+2n+2⇒2B=23+24+...+2n+2

⇒2B−B=2n+2−22⇒B=2n+2−22⇒2B−B=2n+2−22⇒B=2n+2−22

⇒A=22−2n+2+22+(n+1).2n+1⇒A=22−2n+2+22+(n+1).2n+1

⇒A=(n+1).2n+1−2n+2⇒A=(n+1).2n+1−2n+2

⇒A=2n+1(n+1−2)⇒A=2n+1(n+1−2)

⇒A=(n−1).2n+1=2(n−1).2n⇒A=(n−1).2n+1=2(n−1).2n

Mà A=2(n−1).2n=2n+10A=2(n−1).2n=2n+10

⇒2(n+1)=210⇒n−1=29⇒2(n+1)=210⇒n−1=29

⇒n−1=512⇒n=513⇒n−1=512⇒n=513

Vậy n=513

A = 2.2² + 3.2³ + 4.2⁴ + ... + n.2ⁿ 

2.A = 2.2³ + 3.2⁴ + 4.2⁵ + ...+ n.2ⁿ⁺¹

=> A - 2.A = 2.2² + (3.2³ - 2.2³)  + (4.2⁴ - 3.2⁴) + ...+ (n - n + 1).2ⁿ - n.2ⁿ⁺¹

=> A = 2.2² + 2³ + 2⁴ + ..+ 2ⁿ - n.2^{n+ 1}  = 2² + (2² + 2³ + ....+ 2^{n+ 1}) - (n+1).2ⁿ⁺¹

=> A =  - 2² -  (2² + 2³ + ....+ 2^{n+ 1}) + (n+1).2ⁿ⁺¹

Tính B = 2² + 2³ + ....+ 2^{n+ 1} => 2.B =  2³ + ....+ 2^{n+ 1} + 2ⁿ⁺² => 2B - B = 2ⁿ⁺² - 2² => B = 2ⁿ⁺² - 2²

Vậy A = 2² - 2ⁿ⁺² + 2² + (n+1).2ⁿ⁺¹ = (n+1).2ⁿ⁺¹ - 2^{n+ 2} = 2ⁿ⁺¹.(n + 1 - 2) = (n-1).2ⁿ⁺¹ = 2(n-1).2ⁿ

Theo bài cho  A = 2(n-1).2ⁿ = 2ⁿ⁺¹⁰ => 2(n - 1) = 2¹⁰ => n - 1 = 2⁹  = 512 => n = 513

Vậy.............

n= 513, tui chỉ biết đáp án nhưng không biết cách làm

đặt A=2+2^2+2^3+...+2^n

     2A=2^2+2^3+2^4+...+2^n+1 (1)

  2A-A=2\(^{n+1}\)-2

     A=2\(^{n+1}\)-2  (2)

từ (1)(2) =>2 + 2\(^2\)+2\(^3\)+...+2\(^n\)=2\(^{n+1}\)-2

                      2\(^2\)+2\(^3\)+...+2\(^n\)=2\(^{n-1}\)-2\(^2\)

                              ..............................

                                             2\(^n\)=2\(^{n-1}\)-2\(^n\)

cộng vế với vế ta có 

 2+2.2\(^2\)+3.2\(^3\)+...+n.2\(^n\)= n.2\(^{n+1}\)- (2+2\(^2\)+2\(^3\)+...+2\(^n\))

2+(2.2\(^2\)+3.2\(^3\)+...+n.2\(^n\)=n.2\(^{n+1}\)- A

     2+2\(^{n+10}\)=n.2\(^{n+1}\)-2\(^{n+1}\)+2

            2\(^{n+10}\)=2\(^{n+1}\).(n-1)

             2\(^{n+1}\). 2\(^9\)=2\(^{n+1}\).(n-1)

=>n-1=2\(^9\)

=>n=2^9+1=513

vậy n=513