\(2+2^2+2^3+2^4+...\)\(+2^{29}+2^{30}\)
=\(2+\left(2.2\right)+\left(2.2^2\right)+\left(2.2^3\right)+...+\left(2.2^{28}\right)+\left(2.2^{29}\right)\)
=2.(1+2+22+23+...+228+229)
=2.(1+1073741822)
=2.1073741823
=2147483646
Ta có:
\(2+2^2+2^3+2^4+...+2^{29}+2^{30}\)
\(=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{29}+2^{30}\right)\)
\(=2.\left(1+2\right)+2^3.\left(1+2\right)+...+2^{29}.\left(1+2\right)\)
\(=2.3+2^3.3+...+2^{29}.3\)
\(\Rightarrow\left(2+2^2+2^3+...+2^{30}\right)⋮3\)
Ta lại có:
\(2+2^2+2^3+2^4+...+2^{29}+2^{30}\)
\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{28}+2^{29}+2^{30}\right)\)
\(=2.\left(1+2+2^2\right)+2^4.\left(1+2+2^2\right)+...+2^{28}.\left(1+2+2^2\right)\)
\(=2.7+2^4.7+...+2^{28}.7\)
\(\Rightarrow\left(2+2^2+2^3+2^4+...+2^{29}+2^{30}\right)⋮7\)
