\(a.S=2+2^2+2^3+...+2^{20}\\2S=2^2+2^3+...+2^{21}\\ 2S-S=\left(2^2+2^3+...+2^{21}\right)-\left(2+2^2+2^3+...+2^{20}\right)\\ S=2^{21}-2\\ b,A=5+5^2+5^3+...+5^{96}\\ 5A=5^2+5^3+5^4+.......+5^{97}\\ 5A-A=\left(5^2+5^3+...+5^{97}\right)-\left(5+5^2+5^3+...+5^{96}\right)\\ 4A=5^{97}-5\\ A=\dfrac{5^{97}-5}{4}\)
\(S=2+2^2+2^3+...+2^{20}\)
\(\Rightarrow S=2\left(1+2^1+2^2+...+2^{19}\right)\)
\(\Rightarrow S=2.\dfrac{2^{19+1}-1}{2-1}=2\left(2^{20}-1\right)\)
\(B=5+5^2+5^3+...+5^{96}\)
\(\Rightarrow B=5\left(1+5^1+5^2+...+5^{95}\right)\)
\(\Rightarrow B=5.\dfrac{5^{95+1}-1}{5-1}=\dfrac{5\left(5^{96}-1\right)}{4}\)
