Question

CMR: Với mọi n nguyên dương thì \(^{3^{n+2}}\)-\(^{2^{n+2}}\)+\(^{3^n}\)-\(^{2^n}\) chia hết cho 10

Answer 1

\(3^{n+2}-2^{n+2}+3^n-2^n\)

\(=3^n\left(3^2+1\right)-2^{n-1}\left(2^3+2\right)\)

\(=3^n\cdot10-2^{n-1}\cdot10\)

\(=10\left(3^n-2^{n-1}\right)⋮10\forall n\)

Answer 2

3ⁿ⁺²-2ⁿ⁺²+3ⁿ-2ⁿ

=(3ⁿ⁺²+3ⁿ)+(-2ⁿ⁺²-2ⁿ)

=3ⁿ.(3²+1)-2ⁿ.(2²+1)

=3ⁿ.10-2ⁿ.5

=3ⁿ.10-2^n-1.10

=10.(3ⁿ-2^n-1) chia hết cho 10

Vậy 3ⁿ⁺²-2ⁿ⁺²+3ⁿ-2ⁿ chia hết cho 10