Question

Cho A=4 +2²+ 2³+ 2⁴+ .......+2²⁰.

 a .cmr A chia hết 2²⁰   

b)  tìm x€N để A=2^11x-1

Answer 1

\(A=2^2+2^2+2^3+.....+2^{20}\) 

\(2A=2^3+2^3+2^4+...+2^{21}\)

\(\Rightarrow2A-A=2^3+2^{21}-2^2-2^2=8+2^{21}-8=2^{21}\)

\(\Rightarrow A=2^{21}\div2^{20}=2\)

b) \(A=2^{11x-1}\Leftrightarrow2^{2x-1}=2^{21}\Leftrightarrow2x-1=21\Rightarrow2x=21+1=22\Rightarrow x=22\div2=11\)

Answer 2

a) A = 4 + 2² + 2³ + 2⁴ + ... + 2²⁰

Đặt B = 2² + 2³ + 2⁴ + ... + 2²⁰

2B = 2( 2² + 2³ + 2⁴ + ... + 2²⁰ )

= 2³ + 2⁴ + 2⁵ + ... + 2²¹

2B - B = (2³ + 2⁴ + 2⁵ + ... + 2²¹) - (2² + 2³ + 2⁴ + ... + 2²⁰)

B = 2²¹ - 2² = 2²¹ - 4

=> A = 4 + 2²¹ - 4 = 2²¹ = 2²⁰.2 chia hết cho 2²⁰

=> A chia hết cho 2²⁰ ( ĐPCM)

Answer 3

Theo a ) A = 2²¹ 

Mà theo đề bài A = 2^11x-1

=> A = 2²¹ = 2^11x-1

\(\Leftrightarrow21=11x-1\)

\(\Leftrightarrow22=11x\)

\(\Rightarrow x=2\)

Vậy x = 2