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/ \(C=2+2^2+2^3+--..+2^{99}+2^{100}\)
\(\Leftrightarrow2C=2^2+2^3+-...+2^{100}+2^{101}\)
\(\Leftrightarrow2C-C=\left(2^2+2^3+-..+2^{101}\right)-\left(2+2^2+...+2^{100}\right)\)
\(\Leftrightarrow C=2^{101}-2\)
\(2^{2x-1}-2=C\)
\(\Leftrightarrow2^{2x-1}-2=2^{101}-2\)
\(\Leftrightarrow2^{2x}=2^{101}\)
\(\Leftrightarrow2x=101\)
\(\Leftrightarrow x=\dfrac{101}{2}\)
Vậy ..