Đặt A=\(2.2^2+3.2^2+...+n.2^n\)

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

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

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

Đặt \(B=2^3+2^4+...+2^n\Rightarrow2B=2^4+2^5+...+2^{n+1}\)

\(\Rightarrow2B-B=\left(2^4+2^5+...+2^{n+1}\right)-\left(2^3+2^4+...+2^n\right)\)

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

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

=\(n.2^{n+1}-8-2^{n+1}+8=\left(n-1\right).2^{n+1}\)

Mà A=\(2^{n+11}\Rightarrow\left(n-1\right).2^{n+1}=2^{n+11}\)

\(\Rightarrow2^{n+1}.\left(n-1\right)=2^{n+1}.2^{10}\Rightarrow n-1=2^{10}=1024\Rightarrow n=2015\)

Vậy...

p hk tốt nha

Đặt \(A=2.2^2+3.2^3+4.2^4+...+n.2^n\)

Ta có:

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

\(\Rightarrow2A=2\left(2.2^2+3.2^3+4.2^4+...+n.2^n\right)\)

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

\(\Rightarrow2A-A=2.2^2+\left(3.2^3-2.2^3\right)+...+\left(n-n+1\right).2^n-n.2^{n+1}\)

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

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

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

Đặt \(B=2^2+2^3+...+2^{n+1}\)

\(\Rightarrow2B=2^3+2^4+...+2^{n+2}\)

\(\Rightarrow2B-B=2^{n+2}-2^2\Rightarrow B=2^{n+2}-2^2\)

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

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

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

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

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

\(\Rightarrow2\left(n+1\right)=2^{10}\Rightarrow n-1=2^9\)

\(\Rightarrow n-1=512\Rightarrow n=513\)

Vậy \(n=513\)

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Câu hỏi của Thái Hoàng Thục Anh - Toán lớp 7 - Học toán với OnlineMath

help tớ với, chủ nhật phải kt luôn r 

2.2² + 3.2³ + 4.2⁴ + ... + n.2ⁿ  = 2ⁿ⁺¹¹

Đặt vế trái là A ta có:

    A = 2.2² + 3.2³ + 4.2⁴ + ... (n -1).2^n-1+ n.2ⁿ 

2A =  2.2³ + 3.2⁴ + 4.2⁵ +....+ (n- 1).2ⁿ + n.2ⁿ⁺¹

2A-A = [2.2³+3.2⁴ + 4.2⁵ +...+(n-1).2ⁿ+n.2ⁿ⁺¹] - [2.2² + 3.2³+...+n.2ⁿ]

A   = -2.2²+ (2.3³-3.2³) + (3.2⁴ - 4.2⁴) +...+ [(n-1).2ⁿ - n.2ⁿ ] + n.2ⁿ⁺¹

A = -2.2² - 2³ - 2⁴ -...- 2ⁿ + n.2ⁿ⁺¹

Đặt   B = -2.2² - 2³ - 2⁴ - ... - 2ⁿ 

     2B  = -2.2³ - 2⁴ - 2⁵ -...-2ⁿ⁺¹

2B - B = (-2.2³ - 2⁴ - 2⁵ -..-2ⁿ⁺¹) - (-2.2²-2³-2⁴-..-2ⁿ)

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

B = (-2⁴ + 2³) + (- 2ⁿ⁺¹ + 2³) +(-2⁴+2⁴)+(-2⁵+2⁵)+(-2ⁿ+2ⁿ)

B = -16 + 8 - 2ⁿ⁺¹ + 8

B = (-16 + 8 + 8 ) - 2ⁿ⁺¹

B = - 2ⁿ⁺¹

A = n.2ⁿ⁺¹ - 2ⁿ⁺¹

Theo bài ra ta có: 

n.2ⁿ⁺¹ - 2ⁿ⁺¹ = 2ⁿ⁺¹¹

n.2ⁿ⁺¹ - 2ⁿ⁺¹ - 2ⁿ⁺¹¹ = 0

2ⁿ⁺¹.(n - 1 - 2¹⁰) = 0

Vì n là số tự nhiên nên 2ⁿ⁺¹ > 0

Vậy 2ⁿ⁺¹.(n - 1- 2¹⁰) = 0 ⇔ n - 1 - 2¹⁰ = 0 ⇒ n = 1 + 2¹⁰ ⇒ n =  1025

Vậy n = 1025

Đặt \(A=2.2^2+3.2^3+4.2^4+...+n.2^n\)

\(\Leftrightarrow2A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}\)

\(\Leftrightarrow2A-A=2.2^3+3.2^4+4.2^5+...+n.2^{n+1}-\left(2.2^2+3.2^3+4.2^4+...+n.2^n\right)\)

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

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

mà \(A=2^{n+11}\) \(\Leftrightarrow\left(n-1\right).2^{n+1}=2^{n+11}\)

\(\Leftrightarrow\left(n-1\right).2^n.2=2^n.2^{11}\)

\(\Leftrightarrow\left(n-1\right)=2^{10}\)

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

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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