+ Với \(n=1\Rightarrow\left(7^n+1\right)\left(7^n+2\right)=8.9⋮3\)
+ Giả sử có \(A=\left(7^k+1\right)\left(7^k+2\right)=7^{2k}+3.7^k+2⋮3\) Ta cần c/m \(B=\left(7^{k+1}+1\right)\left(7^{k+1}+2\right)⋮3\)
Ta có
\(B=7^{2k+2}+3.7^{k+1}+2=7^2.7^{2k}+3.7.7^k+2\)
\(B=\left(7^{2k}+3.7^k+2\right)+48.7^{2k}+18.7^k=A+3\left(16.7^{2k}+6.7^k\right)\)
Ta có \(A⋮3;3\left(16.7^{2k}+6.7^k\right)⋮3\Rightarrow B⋮3\)
\(\Rightarrow\left(7^n+1\right)\left(7^n+2\right)⋮3\forall n\)
(Dùng phương pháp quy nạp)