Question

\(\left(1+2+2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^9+2^{10}+2^{11}\right)⋮9\)

Answer 1

Đặt A = 1 + 2 + 2² + 2³ + 2⁴ + ... + 2¹¹

2A = 2 + 2² + 2³ + 2⁴ + 2⁵ + ... + 2¹²

2A - A = (2 + 2² + 2³ + 2⁴ + 2⁵ + ... + 2¹²) - (1 + 2 + 2² + 2³ + 2⁴ + ... + 2¹¹)

A = 2¹² - 1

Ta có: \(2^6\equiv1\left(mod9\right)\)

=> \(2^{12}\equiv1\left(mod9\right)\)

=> \(2^{12}-1⋮9\)

=> \(A⋮9\left(đpcm\right)\)

Answer 2

Đặt A = 1 + 2 + 2² + 2³ + 2⁴ + 2⁵ + 2⁶ + 2⁷ + 2⁸ + 2⁹ + 2¹⁰ + 2¹¹

A = (1 + 2³) + (2 + 2⁴) + (2² + 2⁵) + (2⁶ + 2⁹) + (2⁷ + 2¹⁰) + (2⁸ + 2¹¹)

A = 9 + 2.(1 + 2³) + 2².(1 + 2³) + 2⁶.(1 + 2³) + 2⁷.(1 + 2³) + 2⁸.(1 + 2³)

A = 9 + 2.9 + 2².9 + 2⁶.9 + 2⁷.9 + 2⁸.9

A = 9 .(1 + 2 + 2² + 2⁶ + 2⁷ + 2⁸) \(⋮9\left(đpcm\right)\)