Question

Chứng minh rằng 2²ⁿ(2²ⁿ⁺¹-1)-1 chia hết cho 9 với mọi n>1

Answer 1

2²ⁿ(2²ⁿ+1-1)-1

\(=2^{4n+1}-2^{2n}-1=2.2^{4n}-2^{2n}-1\)

\(=2\left(2^{2n}\right)^2-2^{2n}-1=A\)

Đặt \(2^{2n}=t\)

\(\Rightarrow A=2t^2-t-1=\left(2t+1\right)\left(t-1\right)\)

\(=\left(2.2^{2n}+1\right)\left(2^{2n}-1\right)\)

\(=\left(2^{2n+1}+1\right)\left(2^{2n}-1\right)=\left(2+1\right)\left(2^{2n}-2^{2n-1}+...+1\right)\left(2+1\right)\left(2^{2n-1}+...-1\right)\)

\(=9.B\)

Vậy \(A⋮9\)