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![](https://rs.olm.vn/images/avt/0.png?1311)
\(3+3^2+3^3+...+3^{2012}\)
\(=\left(3+3^2+3^3+3^4\right)+...+\left(3^{2009}+3^{2010}+3^{2011}+3^{2012}\right)\)
\(=3\left(1+3+3^2+3^3\right)+...+3^{2009}\left(1+3+3^2+3^3\right)\)
\(=40\left(3+...+3^{2009}\right)⋮40\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
vì 3^1 chia hết cho3
3^2 chia hết cho 3
.....
3^60 chia hết cho 3
mà ta có tính chất :a chia hết cho c
b chia hết cho c
(a+b) chia hết cho c
nên tổng trên chia hết cho 3
Dùng kí hiệu chia hết nha:)
còn chia hết cho 4 thì:
3^1+3^2+....+3^60
=(3^1+3^2)+(3^3+3^4)+....+(3^59+3^60)
=12+3^2 x (3+3^2)+.....+3^58 x (3+3^2)
=12+3^2 x 12+....+3^58 x 12
=12 x (3^2 +......+3^58)
=4 x 3 x (3^2+...+3^58) chia hết cho 4
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=3+3^2+3^3+...+3^{20}\)
\(A=\left(3+3^2\right)+\left(3^3+3^4\right)+...+\left(3^{19}+3^{20}\right)\)
\(A=3\left(1+3\right)+3^3\left(3+1\right)+...+3^{19}\left(1+3\right)\)
\(\Rightarrow A=4\left(3+3^3+...+3^{19}\right)\)
\(\Rightarrow A⋮4\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
a) C = 1 + 3 + 32 + 33 + ... + 311
C = 30 + 3 + 32 + 33 + ... + 311
C = ( 30 + 3 + 32 ) + ( 33 + 34 + 35 ) + ... + ( 39 + 310 + 311 )
C = ( 30 + 3 + 32 ) + 33 . ( 30 + 3 + 32 ) + ... + 39 . ( 30 + 3 + 32 )
C = 13 + 33 . 13 + ... + 39 . 13
C = 13 . ( 1 + 33 + ... + 39 ) \(⋮\) 13 ( đpcm )
b) C = 1 + 3 + 32 + 33 + ... + 311
C = 30 + 3 + 32 + 33 + ... + 311
C = ( 30 + 3 + 32 + 33 ) + ( 34 + 35 + 36 + 37 ) + ( 38 + 39 + 310 + 311 )
C = ( 30 + 3 + 32 + 33 ) + 34 . ( 30 + 3 + 32 + 33 ) + 38 . ( 30 + 3 + 32 + 33 )
C = 40 + 34 . 40 + 38 . 40
C = 40 . ( 1 + 34 + 38 ) \(⋮\) 40 ( đpcm )
c) A = 4 + 42 + 43 + ... + 423 + 424
A = ( 4 + 42 ) + ( 43 + 44 ) + ... + ( 423 + 424 )
A = ( 4 + 42 ) + 42 . ( 4 + 42 ) + ... + 422 . ( 4 + 42 )
A = 20 + 42 . 20 + ... + 422 . 20
A = 20 . ( 1 + 42 + ... + 422 ) \(⋮\) 20 ( đpcm )
d) A = 4 + 42 + 43 + ...+ 423 + 424
A = ( 4 + 42 + 43 ) + ( 44 + 45 + 46 ) + .... + ( 422 + 423 + 424 )
A = ( 4 + 42 + 43 ) + 43 . ( 4 + 42 + 43 ) + ... + 421 . ( 4 + 42 + 43 )
A = 84 + 43 . 84 + ... + 421 . 84
A = 84 . ( 1 + 43 + ... + 421 )
Vì 81 \(⋮\) 9
=> A = 84 . ( 1 +43 + ... + 421 ) \(⋮\) 21 ( đpcm )
e) A = 4 + 42 + 43 + ... + 423 + 424
A = ( 4 + 42 + 43 + 44 + 45 + 46 ) + ... + ( 417 + 418 + 419 + 421 + 422 + 423 + 424 )
A = ( 4 + 42 + 43 + 44 + 45 + 46 ) + ...+ 416 . ( 4 + 42 + 43 + 44 + 45 + 46 )
A = 5460 + ... + 416 . 5460
A = 5460 . ( 1 + ... + 416 )
Vì 5460 \(⋮\) 420
=> A = 5460 . ( 1 + ... + 416 ) \(⋮\) 420 ( đpcm )
Giải:
*A = 4 + 42 + 43 + ... + 423 + 424
A = (4 + 42) + (43 + 44) + ... + (423 + 424)
A = 1 . (4 + 42) + 42 . (4 + 42) + ... + 422 . (4 + 42)
A = 1 . 20 + 42 . 20 + ... + 422 . 20
A = 20 . (1 + 42 + ... + 422)
Vì 20 \(⋮\)20 nên suy ra 20 . (1 + 42 + ... + 422) \(⋮\)20
=> A \(⋮\)20
Vậy A \(⋮\)20
*A = 4 + 42 + 43 + ... + 423 + 424
A = (4 + 42 + 43) + (44 + 45 + 46) + ... + (422 + 423 + 424)
A = 4 . (1 + 4 + 42) + 44 . (1 + 4 + 42) + ... + 422 . (1 + 4 + 42)
A = 4 . 21 + 44 . 21 + ... + 422 . 21
A = 21 . (4 + 44 + ... + 422)
Vì 21\(⋮\)21 nên suy ra 21 . (4 + 44 + ... + 422) \(⋮\)21
=> A \(⋮\)21
Vậy A \(⋮\)21
*A = 4 + 42 + 43 + ... + 423 + 424
A = (4 + 42 + 43 + 44 + 45 + 46) + (47 + 48 + 49 + 410 + 411 + 412) + ... + (419 + 420 + 421 + 422 + 423 + 424)
A = 1 . (4 + 42 + 43 + 44 + 45 + 46) + 46 . (4 + 42 + 43 + 44 + 45 + 46) + ... + 418 . (4 + 42 + 43 + 44 + 45 + 46)
A = 1 . 5460 + 46 . 5460 + ... + 418 . 5460
A = 5460 . (1 + 46 + ... + 418)
Vì 5460 \(⋮\)420 nên suy ra 5460 . (1 + 46 + ... + 418) \(⋮\)420
=> A \(⋮\)420
Vậy A \(⋮\)420.
Chúc bạn học tốt!
![](https://rs.olm.vn/images/avt/0.png?1311)
S=(1+2+2^2+2^3+2^4+2^5)+(2^6+2^7+2^8+2^9+2^10+2^11)
S=63.2^6(1+2+2^2+2^3+2^4+2^5)
S=63+2^6.63
S=63.(1+2^6)
S=63.65:hết cho 9
\(\Rightarrow\) đpcm
1+3+3^2|+......+3^11
=(1+3+3^2+3^3)+(3^4+3^5+3^6+^7)+(3^8+3^9+3^10+3^11)
=40+3^4.40+3^8.40
=40.(1+3^4+3^8) chia hết cho 40