Giải:
\(A=2+22+222+...+22...2\left(100cs\right)\)
\(\Leftrightarrow A=2\left(1+11+111+...+11...1\left(100cs\right)\right)\)
\(\Leftrightarrow A=2.\dfrac{\left(9+99+999+...+99...9\right)}{9}\)
\(\Leftrightarrow A=2.\dfrac{\left(10-1+10^2-1+10^3-1+...+10^{100}-1\right)}{9}\)
\(\Leftrightarrow A=2.\dfrac{\left(10+10^2+10^3+...+10^{100}\right)-100}{9}\)
\(\Leftrightarrow A=2.\dfrac{\left(1+10+10^2+10^3+...+10^{100}-1\right)-100}{9}\)
\(\Leftrightarrow A=2.\dfrac{\left(1+10+10^2+10^3+...+10^{100}\right)-101}{9}\)
\(\Leftrightarrow A=2.\dfrac{\dfrac{10^{101}-1}{9}-101}{9}\)
\(\Leftrightarrow A=2.\dfrac{\dfrac{10^{101}-1-909}{9}}{9}\)
\(\Leftrightarrow A=2.\dfrac{10^{101}-910}{18}\)
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