`~rai~`

$\begin{array}{1}a)S=3+3^2+3^3+...+3^{2019}+3^{2020}\\\Leftrightarrow 3S=3^2+3^3+3^4+...+3^{2020}+3^{2021}\\\Leftrightarrow 3S-S=(3^2+3^3+3^4+...+3^{2020}+3^{2021})-(3+3^2+3^3+...+3^{2019}+3^{2020})\\\Leftrightarrow 2S=3^{2021}-3\\\Leftrightarrow S=\dfrac{3^{2021}-3}{2}\\b)S=3+3^2+3^3+...+3^{2019}+3^{2020}\\S=(3+3^2+3^3+3^4)+(3^5+3^6+3^7+3^8)+...+(3^{2017}+3^{2018}+3^{2019}+3^{2020})\\S=3.(1+3+3^2+3^3)+3^5.(1+3+3^2+3^3)+...+3^{2017}.(1+3+3^2+3^3)\\S=3.40+3^5.40+...+3^{2017}.40\\S=(3+3^5+...+3^{2017}).40 \quad \vdots 40\\\text{Mà}\quad 40\quad \vdots 4;40\quad \vdots 10\\\Rightarrow S\quad \vdots 4;S\quad \vdots 10(đpcm)\end{array}$

a )   S =3¹+3²+....+3 mũ 2019 +3 mũ 2020

=>3S = 3 . ( 3¹+3²+....+3 mũ 2019 +3 mũ 2020 )

=> 3S = 3²+....+3 mũ 2019 +3 mũ 2021

=> 3S - S =( 3²+....+3 mũ 2019 +3 mũ 2021 ) - ( 3¹+3²+....+3 mũ 2019 +3 mũ 2020 ) 

=>    2S   = 3 mũ 2021 - 3¹

=>     S    = ( 3 mũ 2021 - 3¹ ) : 2 

b ) S = (3¹+3²)+....+(3 mũ 2019) +(3 mũ 2020)

=>S = 3 .(1+3)+3³.(1+3)+...+3 mũ 2009.( 1 + 3 )

=>S = 3 .  4      + 3³ . 4 + 3³ . 4 

=>S = 4 . (3+ 3³+...+3 mũ 2009 ) chia hết cho 4

=> S chia hết cho 4

     S=(3¹+3²)+....+(3 mũ 2019) +(3 mũ 2020)

=>S= 3¹.(1+3²) + 3³.(1.3² ) + ...+3 mũ 2019.(1+3²)

=>S= 3¹.  10 + 3³. 10 +...+3 mũ 2019 . 10

=>S=10 . (3¹+3³+...+3 mũ 2019) chia hết cho 10  

=>S chia hết cho 10