Question

chung minh A= 2 + 2^2 +2^3 +2^4 +.........+2^60 chia het cho 7

tim so tu nhien n de : n+4 chia het cho n+1

chung minh ( 1+2 +2^2 +2^3+2^4+2^5+2^6+2^7) chia het cho 3

Answer 1

1. A = 2 + 2² + 2³ + 2⁴ + ... + 2⁶⁰

A = ( 2 + 2² + 2³ ) + ( 2⁴ + 2⁵ + 2⁶ ) + ... + ( 2⁵⁸ + 2⁵⁹ + 2⁶⁰ )

A = 2 ( 1 + 2 + 2² ) + 2⁴ ( 1 + 2 + 2² ) + ... + 2⁵⁸ ( 1 + 2 + 2² )

A = 2 . 7 + 2⁴ . 7 + ... + 2⁵⁸ . 7

A = ( 2 + 2⁴ + ... + 2⁵⁸ ) . 7 => A \(⋮\)7

Vậy ...

2.Ta có : \(n+4⋮n+1\)

Mà : \(n+1⋮n+1\)

\(\Rightarrow\left(n+4\right)-\left(n+1\right)⋮n+1\Rightarrow n+4-n-1⋮n+1\)

\(\Rightarrow3⋮n+1\Rightarrow n+1\in\left\{1;3\right\}\)

\(\Rightarrow n\in\left\{0;2\right\}\)

3. Đặt B = 1 + 2 + 2² + 2³ + 2⁴ + 2⁵ + 2⁶ + 2⁷

B = ( 1 + 2 ) + ( 2² + 2³ ) + ( 2⁴ + 2⁵ ) + ( 2⁶ + 2⁷ )

B = ( 1 + 2 ) + 2² ( 1 + 2 ) + 2⁴ ( 1 + 2 ) + 2⁶ ( 1 + 2 )

B = 1 . 3 + 2² . 3 + 2⁴ . 3 + 2⁶ . 3

B = ( 1 + 2² + 2⁴ + 2⁶ ) . 3 \(\Rightarrow\) B \(⋮\)3

Vậy ...