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 .

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?

Đặt \(A=2.2^2+3.2^3+4.2^4+...+n.2^n\)

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

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

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

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

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

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

Đặt \(B=2^3+2^4+...+2^n\)

=>\(2.B=2^4+2^5+...+2^{n+1}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

=>\(n-1=512\)

=>\(n=513\)

Vậy n=513

dài thế hình như cô giáo lớp mình giải còn ngắn hơn thế này