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\(A=2+2^2+2^3+...+2^{2016}\)
\(\Rightarrow2A=2\left(2+2^2+2^3+...+2^{2016}\right)\)
\(\Rightarrow2A=2^2+2^3+2^4+...+2^{2017}\)
\(\Rightarrow2A-A=2^2+2^3+...+2^{2017}-2-2^2-2^3-...-2^{2016}\)
\(\Rightarrow A=2^{2017}-2\)
\(B=12+12^2+12^3+...+12^{100}\)
\(\Rightarrow12B=12\left(12+12^2+12^3+12^{100}\right)\)
\(\Rightarrow12B=12^2+12^3+...+12^{101}\)
\(\Rightarrow12B-B=12^2+12^3+...+12^{101}-12-12^2-12^3-...-12^{100}\)
\(\Rightarrow11B=12^{101}-12\)
\(\Rightarrow B=\dfrac{12^{101}-12}{11}\)
\(C=5+5^2+5^3+...+5^{32}\)
\(\Rightarrow5C=5\left(5+5^2+...+5^{32}\right)\)
\(\Rightarrow5C=5^2+5^3+...+5^{33}\)
\(\Rightarrow5C-C=5^2+5^3+...+5^{33}-5-5^2-...-5^{32}\)
\(\Rightarrow4C=5^{33}-5\)
\(\Rightarrow C=\dfrac{5^{33}-5}{4}\)
