a) \(A=2+2^2+2^3+...+2^{30}\)
\(2A=2^2+2^3+...+2^{31}\)
\(2A-A=2^2+2^3+...+2^{31}-2-2^2-...-2^{30}\)
\(A=2^{31}-2\)
\(\Rightarrow2\left(A+2\right)=2\cdot\left(2^{31}-2+2\right)=2\cdot2^{31}=2^{32}\)
\(\Rightarrow2^{32}=2^{2n}\Rightarrow32=2n\Rightarrow n=\dfrac{32}{2}=16\)
b) \(A=2+2^2+...+2^{30}\)
\(A=\left(2+2^2+2^3\right)+...+\left(2^{28}+2^{29}+2^{30}\right)\)
\(A=2\cdot7+2^4\cdot7+...+2^{28}\cdot7\)
\(A=7\cdot\left(2+2^4+...+2^{28}\right)\)
Vậy: A ⋮ 7
a) \(A=2+2^2+2^3+...+2^{29}+2^{30}\)
\(\Rightarrow2A=2^2+2^3+2^4+...+2^{30}+2^{31}\)
\(\Rightarrow A=2A-A\)
\(=\left(2^2+2^3+2^4+...+2^{30}+2^{31}\right)-\left(2+2^2+2^3+...+2^{29}+2^{30}\right)\)
\(=2^{31}-2\)
\(\Rightarrow2\left(A+2\right)=2\left(2^{31}-2+2\right)=2.2^{31}=2^{32}\)
Mà \(2\left(A+2\right)=2^{2n}\)
\(\Rightarrow2^{2n}=2^{32}\)
\(\Rightarrow2n=32\)
\(\Rightarrow n=32:2\)
\(\Rightarrow n=16\)
b) \(A=2+2^2+2^3+...+2^{29}+2^{30}\)
\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{28}+2^{29}+2^{30}\right)\)
\(=2.\left(1+2+2^2\right)+2^4.\left(1+2+2^2\right)+...+2^{28}.\left(1+2+2^2\right)\)
\(=2.7+2^4.7+...+2^{28}.7\)
\(=7.\left(2+2^4+...+2^{28}\right)⋮7\)
Vậy \(A⋮7\)