Nhận xét:

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

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

Tương tự

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

...

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

Cộng vế với vế ta được:

\(2+2\cdot2^2+3\cdot2^3+4\cdot2^4+...+n\cdot2^n=n\cdot2^{n+1}-\left(2+2^2+2^3+...+2^n\right)=n\cdot2^{n+1}-2^{n+1}+2\)

\(\Rightarrow2\cdot2^2+3\cdot2^3+4\cdot2^4+...+n\cdot2^n=\left(n-1\right)\cdot2^{n+1}\)(1)

Theo giả thiết thì VT(1) = 2ⁿ⁺¹⁰. Ta có:

\(2^{n+10}=\left(n-1\right)\cdot2^{n+1}\Leftrightarrow2^{n+1}\cdot2^9=\left(n-1\right)\cdot2^{n+1}\Leftrightarrow n-1=2^9\Leftrightarrow n=2^9+1\)

Vậy, n = 2⁹ + 1.

(Đề bài thì hay mà bạn đánh câu hỏi cẩu thả quá! :D). 

Đinh Thùy Linh ơ đề bài bảo tìm x mà

 Mình làm đc mỗi 1 câu, Thông cảm

7^6+7^5+7^4 chia hết cho 11

= 7^4.2^2+7^4.7+7^4

= 7^4.(2^2+7+1)

= 7^4. 11

Vì tích này có số 11 nên => chia hết cho 7

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

\(\Rightarrow n-1=512\)

\(\Rightarrow n=513\)

c, \(\frac{-32}{-2^n}=4\)

\(\Rightarrow-2^n=-32:4\)

\(\Rightarrow-2^n=-8\)

\(\Rightarrow-2^n=-2^3\Rightarrow n=3\)

d, \(\frac{8}{2^n}=2\)

\(\Rightarrow2^n=8:2\)

\(\Rightarrow2^n=4\)

\(\Rightarrow2^n=2^2\Rightarrow n=2\)

e, \(\frac{25^3}{5^n}=25\)

\(\Rightarrow5^n=25^3:25\)

\(\Rightarrow5^n=25^2\)

\(\Rightarrow5^n=5^4\Rightarrow n=4\)

i , \(8^{10}:2^n=4^5\)

\(\Rightarrow2^n=8^{10}:4^5\)

\(\Rightarrow2^n=\left(2^3\right)^{10}:\left(2^2\right)^5\)

\(\Rightarrow2^n=2^{30}:2^{10}\)

\(\Rightarrow2^n=2^{20}\Rightarrow n=20\)

k, \(2^n.81^4=27^{10}\)

\(\Rightarrow2^n=27^{10}:81^4\)

\(\Rightarrow2^n=\left(3^3\right)^{10}:\left(3^4\right)^4\)

\(\Rightarrow2^n=3^{30}:3^{16}\)

\(\Rightarrow2^n=3^{14}\)

\(\Rightarrow2^n=4782969\)Không chia hết cho 2 nên ko có Gt n thỏa mãn