1) \(\Rightarrow2P=2^2+2^3+...+2^{101}\)
\(\Rightarrow P=2P-P=2^2+2^3+...+2^{101}-2-2^2-...-2^{100}=2^{101}-2\)
2) \(\Rightarrow3A=3+3^2+...+3^{101}\)
\(\Rightarrow2A=3A-A=3+3^2+...+3^{101}-1-3-...-3^{100}=3^{101}-1\)
\(\Rightarrow A=\dfrac{3^{101}-1}{2}\)
3) \(\Rightarrow2B=2-2^2+...-2^{100}+2^{101}\)
\(\Rightarrow3B=2B+B=2-2^2+...-2^{100}+2^{101}+1-2+2^2-....+2^{100}=2^{101}+1\)
\(\Rightarrow B=\dfrac{2^{101}+1}{2}\)
4) \(Q=\left(2.2^2.2^3.2^4...2^{10}\right):2^{52}=2^{\dfrac{\left(10+1\right)\left(10-1+1\right)}{2}}:2^{52}=2^{55}:2^{52}=2^3=8\)
\(\Rightarrow P=2P-P=2^{101}+2^{100}+...+2^3+2^2-2^{100}-2^{99}-...-2^2-2\\ \Rightarrow P=2^{101}-2\\ \Rightarrow2A=3A-A=3+3^2+...+3^{101}-1-3-...-3^{100}\\ \Rightarrow2A=3^{101}-1\Rightarrow A=\dfrac{3^{101}-1}{2}\\ \Rightarrow3B=2B+B=2^{101}-2^{100}+...+2^3-2^2+2+2^{100}-2^{99}+2^{98}-...-2+1\\ \Rightarrow3B=2^{101}+1\Rightarrow B=\dfrac{2^{101}+1}{3}\)
\(Q=2\cdot2^2\cdot...\cdot2^{10}=2^{1+2+...+10}=2^{55}:2^{52}=2^3=8\)
