\(A=2+2^2+2^3+...........+2^{20}\)
\(\Leftrightarrow2A=2^2+2^3+............+2^{20}+2^{21}\)
\(\Leftrightarrow2A-A=\left(2^2+2^3+........+2^{21}\right)-\left(2+2^2+.....+2^{20}\right)\)
\(\Leftrightarrow A=2^{21}-2\)
A = 2 + 22 + 23 + ..... + 220
\(\Rightarrow\) 2A = 22 + 23 + 24 + .......... + 221
\(\Rightarrow\) 2A - A = ( 22 + 23 + 24 + ..... + 221 ) - ( 2 + 22 + 23 + ........ + 220 )
\(\Rightarrow\) 2A - A = 22 + 23 + 24 + ....... + 221 - 2 - 22 - 23 - ........... - 220
\(\Rightarrow\) A = 221 - 2