Question

cho N = 3\(^{n+2}\) -2\(^{n+2}\) +3\(^n\) -2\(^n\) (với n là số tự nhiên). CMR N chia hết cho 10

Answer 1

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

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

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

\(\Rightarrow N=\left[3^n\left(3^2+1\right)\right]-\left[2^{n-1}\left(2^3+2\right)\right]\)

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

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

\(\Rightarrow N⋮10\left(đpcm\right)\)

Vậy \(N⋮10\)