1) went

2) to buy

3) have

4) are crossing

4) will be

5) get

6) is looking/ doesn't like

7) did/ buy

a) \(B=\dfrac{3n}{n+1}=\dfrac{3\left(n+1\right)}{n+1}-\dfrac{3}{n+1}=3-\dfrac{3}{n+1}\in Z\)

\(\Rightarrow\dfrac{3}{n+1}\in Z\Rightarrow n+1\inƯ\left(3\right)=\left\{1;-1;3;-3\right\}\Rightarrow n\in\left\{0;-2;2;-4\right\}\)

b) \(\dfrac{6n+5}{2n-1}=\dfrac{3\left(2n-1\right)}{2n-1}+\dfrac{8}{2n-1}=3+\dfrac{8}{2n-1}\in Z\)

\(\Rightarrow\dfrac{8}{2n-1}\in Z\Rightarrow2n-1\inƯ\left(8\right)=\left\{1;-1;2;-2;4;-4;8;-8\right\}\)

Vì \(n\in Z\Rightarrow n\in\left\{1;0\right\}\)

\(5\left(x+2\right)\left(x-2\right)-\left(2x-3\right)^2-x^2+17=5\left(x^2-4\right)-\left(4x^2-12x+9\right)-x^2+17=12x-12\)

a) Ta có: \(17^{20}=\left(17^2\right)^{10}=289^{10}>71^{10}>71^5\)

b) \(3.24^{100}=3.3^{100}.8^{100}=3^{101}.2^{300}\)

\(4^{300}=2^{300}.2^{300}=4^{150}.2^{300}\)

Vì \(3^{101}.2^{300}< 4^{150}.2^{300}\Rightarrow3.24^{100}< 4^{300}\Rightarrow3.24^{100}< 3^{300}+4^{300}\)

\(-x^2+4xy-4y^2=-\left(x-2y\right)^2\)

\(x^3-9x=0\Leftrightarrow x\left(x^2-9\right)=0\Leftrightarrow x\left(x-3\right)\left(x+3\right)=0\Leftrightarrow\left[{}\begin{matrix}x=0\\x-3=0\\x+3=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=0\\x=3\\x=-3\end{matrix}\right.\)

a) Ta có: 2, 2², 2³,...2²⁰ đều chia hết cho 2

=> A=\(2^2+2^3+...+2^{20}⋮2\)

b) \(A=2+2^2+2^3+...+2^{20}=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{19}\left(1+2\right)=3.2+3.2^3+...+3.2^{19}=3\left(2+2^3+...+2^{19}\right)⋮3\)c) \(A=2+2^2+2^3+...+2^{20}=2\left(1+2+2^2+2^3\right)+2^5\left(1+2+2^2+2^3\right)+...2^{17}\left(1+2+2^2+2^3\right)=15\left(2+2^5+...+2^{17}\right)⋮15\Rightarrow⋮5\Rightarrow A⋮5\)

a)\(A=2+2^2+2^3+2^4+...+2^{100}=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{99}\left(1+2\right)=3.2+3.2^3+...+3.2^{99}=3\left(2+2^3+...+2^{99}\right)⋮3\)b) \(B=4+4^2+4^3+...+4^{2022}=4\left(1+4\right)+4^3\left(1+4\right)+...+4^{2021}\left(1+4\right)=5.4+5.4^3+...+5.4^{2021}=5\left(4+4^3+...+4^{2021}\right)⋮5\)

Despite the expensive price, my wife insists on buying the car.