to biet tra loi nhung dai lam khong biet danh mu

21537

bạn nào tl được thì giúp mik vs!mik cũng đang khó câu này?

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

2.2ⁿ+3.2ⁿ+4.2ⁿ+5.2ⁿ+...+n.2ⁿ=2ⁿ⁺¹⁰

<=>(2+3+4+5+...+n).2ⁿ=2¹⁰.2ⁿ

<=>2+3+4+5+...+n=2¹⁰

<=>n không tồn tại

Đặt \(A=2.2^2+3.2^3+4.2^4+5.2^5+...+n.2^n\)

\(\Rightarrow2A=2.2^3+3.2^4+4.2^5+5.2^6+...+n.2^{n+1}\)

\(\Rightarrow2A-A=2.2^3+3.2^4+4.2^5+5.2^6+...+n.2^{n+1}\)

\(-2.2^2-3.2^3-4.2^4-5.2^5-...-n.2^n\)

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

Đặt \(M=\left(2^3+2^4+...+2^n\right)\)

\(\Rightarrow2M=\left(2^4+2^5+...+2^{n+1}\right)\)

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

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

\(\Rightarrow A=\left(n-1\right)2^{n+1}=2^{n+10}\)

\(\Rightarrow\left(n-1\right)=2^9\)

\(\Rightarrow n=513\)

Đặt \(A=2.2^2+3.2^3+4.2^4+...+n.2^n=2^{n+10}\)

\(\Rightarrow2A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}\)

\(\Rightarrow2A-A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}-2.2^2-3.2^3-4.2^4-...-n.2^n\)

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

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

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

\(\Leftrightarrow A=-2^3-2^4+2^3-2^5+2^4-...-2^{n+1}+2^n+n.2^{n+1}\)

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

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

Mà \(A=2^{n+10}=2^{n+1}.2^9=2^{n+1}.512\)

\(\Rightarrow n-1=512\)

\(\Rightarrow n=513\)

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

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

2A - A = (2.2³ - 3.2³) + (3.2⁴ - 4.2⁴) + ... + [(n-1).2ⁿ - n.2ⁿ] + n.2ⁿ⁺¹

A = -2³ - 2⁴ - ... - 2ⁿ + n.2ⁿ⁺¹ - 2.2²

A = n.2ⁿ⁺¹ - (2³ + 2⁴ + 2⁵ + ... + 2ⁿ) - 2³

Đặt B = 2³ + 2⁴ + ... + 2ⁿ

2B = 2⁴ + 2⁵ + ... + 2ⁿ⁺¹

2B - B = 2⁴ + 2⁵ + ... + 2ⁿ⁺¹ - 2³ - 2⁴ - 2ⁿ

B = 2ⁿ⁺¹ - 2³

Mà A = n.2ⁿ⁺¹ - (2³ + 2⁴ + 2⁵ + ... + 2ⁿ) - 2³

=> A = n.2ⁿ⁺¹ - B - 2³

=> A = n.2ⁿ⁺¹ - (2ⁿ⁺¹ - 2³) - 2³

A = n.2ⁿ⁺¹ - 2ⁿ⁺¹ + 2³ - 2³

A = (n-1).2ⁿ⁺¹

Mà 2.2²+ 3.2³ + 4.2⁴ + 5.2⁵ + · · · + n.2ⁿ = 2ⁿ⁺¹⁰

=> A = 2ⁿ⁺¹⁰

=> (n-1).2ⁿ⁺¹ = 2ⁿ⁺¹⁰

(n-1) = 2ⁿ⁺¹⁰ : 2ⁿ⁺¹

n-1 = 2⁹

n = 512 + 1

n = 513

Nhận tớ một lạy Hải ạ!!! :)

Con lậy má .