toán chứng minh dễ mà bn

a)đặt tên biểu thức là C . Ta có :
C =  1 + 4 + 4² + 4³ + ... + 4²⁰¹² 

C = ( 1 + 4 + 4² ) + ( 4³ + 4⁴ + 4⁵ ) + ... + ( 4²⁰¹⁰ + 4²⁰¹¹ + 4²⁰¹² )

C = 21 + 4³ . ( 1 + 4 + 4² ) + ... + 4²⁰¹⁰ . ( 1 + 4 + 4² )

C = 21 + 4³ . 21 + ... + 4²⁰¹⁰ . 21

C = 21 . ( 1 + 4³ + ... + 4²⁰¹⁰ ) 

=> C chia hết cho 21

b) đặt tên biểu thức là B . Ta có :

B =  1 + 7 + 7² + ... + 7¹⁰¹

B = ( 1 + 7 ) + ( 7² + 7³ ) + ... + ( 7¹⁰⁰ + 7¹⁰¹ )

B = 8 + 7² . ( 1 + 7 ) + ... + 7¹⁰⁰. ( 1 + 7 )

B = 8 + 7² . 8 + ... + 7¹⁰⁰ . 8

B = 8 . ( 1 + 7² + ... + 7¹⁰⁰ )

=> B chia hết cho 8

tương tự

1) \(1+4+4^2+4^3+...+4^{2012}\)

\(=\left(1+4+4^2\right)+\left(4^3+4^4+4^5\right)+...+\left(4^{2010}+4^{2011}+4^{2012}\right)\)

\(=21+21\cdot4^3+...+21\cdot4^{2010}\)

\(=21\cdot\left(1+4^3+...+4^{2010}\right)\) chia hết cho 21

2) \(1+7+7^2+7^3+...+7^{101}\)

\(=\left(1+7\right)+\left(7^2+7^3\right)+...+\left(7^{100}+7^{101}\right)\)

\(=8+8\cdot7^2+...8\cdot7^{100}\)

\(=8\cdot\left(1+7^2+...+7^{100}\right)\) chia hết cho 8

3) CM chia hết cho 5:

\(2+2^2+2^3+2^4+...+2^{100}\)

\(=\left(2+2^3\right)+\left(2^2+2^4\right)+...+\left(2^{98}+2^{100}\right)\)

\(=5\cdot2+5\cdot2^2+...+5\cdot2^{98}\)

\(=5\cdot\left(2+2^2+...+2^{98}\right)\) chia hết cho 5

CM chia hết cho 31:

\(2+2^2+2^3+...+2^{100}\)

\(=\left(2+2^2+2^3+2^4+2^5\right)+...+\left(2^{96}+2^{97}+2^{98}+2^{99}+2^{100}\right)\)

\(=2\cdot31+...+2^{96}\cdot31\)

\(=31\cdot\left(2+...+2^{96}\right)\) chia hết cho 31

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Bài 1:
$A=2^1+2^2+2^3+2^4$

$2A=2^2+2^3+2^4+2^5$

$\Rightarrow 2A-A=2^5-2^1$

$\Rightarrow A=2^5-1=32-1=31$

----------------------------

$B=3^1+3^2+3^3+3^4$

$3B=3^2+3^3+3^4+3^5$

$\Rightarrow 3B-B = 3^5-3$

$\Rightarrow 2B = 3^5-3\Rightarrow B = \frac{3^5-3}{2}$

--------------------------

$C=5^1+5^2+5^3+5^4$

$5C=5^2+5^3+5^4+5^5$

$\Rightarrow 5C-C=5^5-5$

$\Rightarrow C=\frac{5^5-5}{4}$

Bài 2: Sai đề bạn nhé. Bạn xem lại.

\(A=\left(1+4+4^2+4^3+4^4\right)+...+4^{95}\left(1+4+4^2+4^3+4^4\right)\)

\(=341\cdot\left(1+...+4^{95}\right)⋮31\)

\(A=\left(2+2^2\right)+...+\left(2^{99}+2^{100}\right)\)

\(A=2\cdot\left(1+2\right)+...+2^{99}\cdot\left(1+2\right)\)

\(A=2\cdot3+...+2^{99}\cdot3\)

\(A=3\cdot\left(2+...+2^{99}\right)⋮3\left(đpcm\right)\)

2 ý kia tương tự

Giải:

Đặt S=(2+2^2+2^3+...+2^100)

=2.(1+2+2^2+2^3+2^4)+2^6.(1+2+2^2+2^3+2^4)+...+(1+2+2^2+2^3+2^4).296

=2.31+26.31+...+296.31

=31.(2+26+...+296)\(⋮\)31

Ta có :

\(A=2+2^2+2^3+2^4+...+2^{99}+2^{100}\)

=> \(A=(2+2^2)+(2^3+2^4)+...+(2^{99}+2^{100})\)

=> \(A=2(1+2)+2^3(1+2)+...+2^{99}(1+2)\)

=> \(A=2.3+2^3.3+...+2^{99}.3\)

=> \(A=(2+2^3+...+2^{99}).3\)chia hết cho 3             ( 1 )

Ta lại có :

\(A=2+2^2+2^3+2^4+...+2^{99}+2^{100}\)

=> \(A=2(1+2+2^2+2^3+...+2^{98}+2^{99})\)chia hết cho 2       ( 2 )

Từ ( 1 ) và ( 2 ) ta có :

A chia hết cho 2 . 3 hay A chia hết cho 6

Ta có :

\(A=2+2^2+2^3+2^4+...+2^{99}+2^{100}\)

=> ​\(A=\left(2+2^2+2^3+2^4+2^5\right)+....\left(2^{96}+2^{97}+2^{98}+2^{99}+2^{100}\right)\)​

=> \(A=2\left(1+2+2^2+2^3+2^4\right)+...+2^{96}\left(1+2+2^2+2^3+2^4\right)\)

=> \(A=2.31+...+2^{96}.31\)

=> \(A=\left(2+...+2^{96}\right)31\)chia hết cho 31

A=1+4+4²+4³+...+4²⁰¹⁴

A=(1+4+4²)+(4³+4⁵+4⁶)+...+(4²⁰¹²+4²⁰¹³+2²⁰¹⁴)

A=21.(1+4³+...+4²⁰¹²)

B=1+7+7²+...+7¹⁰¹

B=(1+7)+(7²+7³)+...+(7¹⁰⁰+7¹⁰¹)

B=8(1+7²+...+7¹⁰⁰)