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\(B=3+3^2+3^3+3^4+...+3^{2009}+3^{2010}\)
\(=\left(3+3^2\right)+\left(3^3+3^4\right)+...+\left(3^{2009}+3^{2010}\right)\)
\(=3\left(1+3\right)+3^3\left(1+3\right)+...+3^{2009}\left(1+3\right)\)
\(=4.\left(3+3^3+...+3^{2009}\right)\)
⇒ \(B\) ⋮ 4
b: \(C=5\left(1+5+5^2\right)+...+5^{2008}\left(1+5+5^2\right)=31\cdot\left(5+...+5^{2008}\right)⋮31\)
\(A=1+3+3^2+3^3+...+3^{102}+3^{103}\)
\(\Rightarrow A=\left(1+3\right)+\left(3^2+3^3\right)+...+\left(3^{102}+3^{103}\right)\)
\(\Rightarrow A=\left(1+3\right)+3^2\left(1+3\right)+...+3^{102}\left(1+3\right)\)
\(\Rightarrow A=\left(1+3\right)\left(1+3^2+...+3^{102}\right)\)
\(\Rightarrow A=4\left(1+3^2+...+3^{102}\right)⋮4\)
\(S=\left(1+3\right)+...+3^8\left(1+3\right)=4\left(1+...+3^8\right)⋮4\)
\(S=\left(1+3+3^2\right)+...+3^7\left(1+3+3^2\right)\)
\(=13\left(1+...+3^7\right)⋮13\)
Phần a sai đề nha
b) S = 3 + 32 + 33 + 34 + ............ + 320
S = ( 3 + 32 ) + ( 33 + 34 ) + ........... + ( 319 + 320 )
S = 3 . ( 1 + 3 ) + 33 . ( 1 + 3 ) + ....... + 319 . ( 1 + 3 )
S = 3 . 4 + 33 . 4 + ............. + 319 . 4
S = 12 + 27 . 4 + ........... + 319 . 4
S = 12 + 108 + ........... + 319 . 4
Mà 12 ; 108 \(⋮\) 12 \(\Rightarrow\) ( 12 + 108 + ............ + 319 . 4 ) \(⋮\) 12
Vậy S \(⋮\) 12 ( ĐPCM )
b/S=3+3^2+3^3+3^4+......+3^20(gồm 21 số hạng)
S=(3+3^2)+(3^3+3^4)+(3^5+3^6)+......+(3^19+3^20)
S=1(3+3^2)+3^2(3+3^2)+......+3^18(3+3^2)
S=1.12 +3^2.12 +........+3^18.12
S=12.(1+3^2+3^4+......+3^18)
Vậy S chia hết cho 12