Question

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

Answer 1

Ta có:\(3^{n+2}-2^{n+2}+3^n-2^n\)

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

=\(3^n.10-2^n.5\)

=\(3^n.10-2^{n-1}.2.5\)

=\(3^n.10-2^{n-1}.10\)

=\(\left(3^n-2^{n-1}\right).10⋮10\)

\(\Rightarrow3^{n+2}-2^{n+2}+3^n-2^n⋮10\)

Nhớ tick cho mình nha!