Question

Chứng minh rằng : \(3^{n+2}-2^{n+2}+3^n-2^n\) chia hết cho 10.

Answer 1

\(3^{n+2}-2^{n+2}+3^n-2^n=3^n \left(3^2+1\right)+2^n\left(2^2+1\right)=3^n.10+2^{n-1}.10=10\left(3^n-2^{n-1}\right)\) chia hết cho 10

Answer 2

ta có

3^n+2=3^n.3^2

2^n-2=2^n.2^2

=>3^n.3^2 - 2^n.2^2 + 3^n-2^n=3^n.(3^2+1) - 2^n.(2^2+1)=3^n.10 - 2^n.5

Mà 3^n.10 luôn chia hết cho 10

2^n.5=10.2^n-1 luôn chia hết cho 10

=>3^n.10-2^n.5 chia hết cho 10

=>3^n+2 - 2^n+2 + 3^2 - 2^n chia hết cho 10