Question

a) CMr : Với mọi số nguyên dương n thì

\(3^{n+2}-2^{n+2}+3^n-2^n⋮10\)

Answer 1

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

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

Answer 2

Ta có \(3^{n+2}-2^{n+2}+3^n-2^n\)

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

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

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

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

Ta thay a là 10; b là \(3^n-2^{n-1}\)

Ta có \(a⋮10\)=>\(a.b⋮10\)

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