$2^{2}$ + $2^{3}$ + $2^{4}$ + ... + $2^{2021}$
= ( $2^{2}$ + $2^{3}$ ) + ... + ( $2^{2020}$ + $2^{2021}$ )
= $2^{2}$ ( 1 + 2 ) + ... + $2^{2020}$ ( 1 + 2 )
= $2^{2}$ . 3 + ... + $2^{2020}$ . 3
= ( $2^{2}$ + ... + $2^{2020}$ ) . 3
--> $2^{2}$ + $2^{3}$ + $2^{4}$ + ... + $2^{2021}$ chia hết cho 3