Question

CMR

3^(2n+1) + 2^(n+2) chia hết cho 7

Answer 1

Ta có: \(3^{\left(2n+1\right)}+2^{n+2}=9^n.3+2^n.4\)

\(=9^n.3-2^n.3+2^n.7\)

\(=3.\left(9^n-2^n\right)+2^n.7\)

Ta có: \(9^n-2^n⋮9-2=7\) , \(2^n.7⋮7\)

\(\Rightarrow3^{\left(2n+1\right)}+2^{\left(n+2\right)}\) chia hết cho 7. ( đpcm )

Answer 2

\(3^{2n+1}+2^{n+2}\)

\(=3^{2n}.3+2^n.2^2\)

\(=\left(3^2\right)^n.3+2^n.4\)

\(=9^n.3+2^n.4\)

\(=9^n.3-2^n.3+2^n.7\)

\(=3\left(9^n-2^n\right)+2^n.7⋮7\)

\(\rightarrowđpcm\)