Question

Cho A = 2 + 2² + 2³ + ... + 2¹¹⁹ + 2¹²⁰ chứng tỏ rằng

a, A chia hết cho 3

b, A chia hết cho 7

Answer 1

A = (2 + 2^2) + (2^3+  2^4)  +...... + (2^119 + 2^120)

A= (2.1+2.2) + (2^3.1 + 2^3.2) + ...... + (2^119.1 + 2^119.2)

A = 2.3 + 2^3.3 + ...... + 2^119.3

A = 3.(2+2^3+......+2^119)

Chia hết cho 3

A = (2 + 2^2 + 2^3)  +...... + (2^118 + 2^119 + 2^120)

A = (2.1 + 2.2 + 2.4) + ....... + (2^118.1 + 2^118.2 + 2^118.4)

A = 2.(1+2+4)  + ...... + 2^118.(1 + 2 + 4)
A=  7.(2 + 2^4 + ...... + 2^118)

Chia hết cho 7