a/ \(C=2+2^2+2^3+......+2^{100}\)
\(=\left(2+2^2+2^3+2^4+2^5\right)+\left(2^6+2^7+2^8+2^9+2^{10}\right)+......+\left(2^{96}+2^{97}+2^{98}+2^{99}+2^{100}\right)\)
\(=2\left(1+2+2^2+2^3+2^4\right)+2^6\left(1+2+2^2+2^3+2^4\right)+......+2^{96}\left(1+2+2^2+2^3+2^4\right)\)
\(=2.31+2^6.31+......+2^{96}.31\)
\(=31\left(2+2^6+.....+2^{96}\right)⋮31\left(đpcm\right)\)
b/ \(C=2+2^2+2^3+..........+2^{99}+2^{100}\)
\(\Leftrightarrow2C=2^2+2^3+.......+2^{100}+2^{101}\)
\(\Leftrightarrow2C-C=\left(2^2+2^3+......+2^{101}\right)-\left(2+2^2+...+2^{100}\right)\)
\(\Leftrightarrow C=2^{101}-2\)
\(2^{2x-1}-2=C\)
\(\Leftrightarrow2^{2x-1}-2=2^{101}-2\)
\(\Leftrightarrow2^{2x}=2^{101}\)
\(\Leftrightarrow2x=101\)
\(\Leftrightarrow x=\dfrac{101}{2}\)
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a) 2.(1+2+4+8+16)+26(1+2+4+8+16)+...+296(1+2+4+8+16)
= 2.31+26.31+...+296.31=31.(2+26+..+296)⋮31
b)
